Keller–Segel–(Navier–)Stokes System Modeling Coral Fertilization

Abstract

Chemotaxis, the directed movement caused by the concentration of certain chemicals, is ubiquitous in biology and ecology, and has a significant effect on pattern formation in numerous biological contexts (Hillen and Painter 2009; Maini et al. 1991).

4.1 Introduction

Chemotaxis, the directed movement caused by the concentration of certain chemicals, is ubiquitous in biology and ecology, and has a significant effect on pattern formation in numerous biological contexts (Hillen and Painter 2009; Maini et al. 1991). The first mathematically rigorous studies of chemotaxis were carried out by Patlak (1953) and Keller–Segel (1970). The latter work involves the derivation of a system of PDEs, now known as the Keller–Segel system, which, despite its simple structure, was proved to have a lasting impact as a theoretical framework describing the collective behavior of populations under the influence of a chemotactic signal produced by the populations themselves (Bellomo et al. 2015; Herrero and Velázquez 1997; Winkler 2013, 2014c). In contrast to this well-understood Keller–Segel system, there seem to be few theoretical results on nontrivial behavior in situations where the signal is not produced by the population, such as in oxygenotaxis processes of swimming aerobic bacteria (Tuval et al. 2005), or where the signal production occurs by indirect processes, such as in glycolysis reaction, tumor invasion and the spread of the mountain pine beetle (Chaplain and Lolas 2005; Dillon et al. 1994; Fujie and Senba 2017; Hu and Tao 2016; Painter et al. 2000; Tao and Winkler 2017b).

In this chapter, we study the decay property of the chemotaxis–fluid systems modeling coral fertilization. Section 4.3 is concerned with the following Keller–Segel–Stokes system

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t+u\cdot \nabla \rho =\varDelta \rho -\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)-\rho m,&(x,t)\in \varOmega \times (0,T), \\&m_t+u\cdot \nabla m=\varDelta m-\rho m,&(x,t)\in \varOmega \times (0,T), \\&c_t+u\cdot \nabla c=\varDelta c-c+m,&(x,t)\in \varOmega \times (0,T), \\&u_t=\varDelta u-\nabla P+(\rho +m)\nabla \phi ,\quad \nabla \cdot u=0,&(x,t)\in \varOmega \times (0,T), \\&(\nabla \rho -\rho \mathscr {S}(x,\rho ,c)\nabla c)\cdot \nu =\nabla m\cdot \nu =\nabla c\cdot \nu =0, u=0,&(x,t)\in \partial \varOmega \times (0,T),\\&\rho (x,0)=\rho _0(x),\quad m(x,0)=m_0(x),\quad c(x,0)=c_0(x),&u(x,0)=u_0(x),\; x\in \varOmega , \end{aligned}\right. \end{aligned}$$

(4.1.1)

where $T\in (0,\infty ]$, $\varOmega \subset \mathbb R^3$ is a bounded domain with smooth boundary $\partial \varOmega$, the chemotactic sensitivity tensor $\mathscr {S}(x,\rho ,c)=(s_{ij}(x,\rho ,c))\in C^2(\overline{\varOmega }\times [0,\infty )^2)$, $i,j\in \{1,2,3\}$, and $\phi \in W^{2,\infty }(\varOmega )$.

This PDE system describes the phenomenon of coral broadcast spawning (Espejo and Suzuki 2017; Espejo and Winkler 2018; Kiselev and Ryzhik 2012a, b), where the sperm $\rho$ chemotactically moves toward the higher concentration of the chemical c released by the egg m, while the egg m is merely affected by random diffusion, fluid transport and degradation upon contact with the sperm. Meanwhile, the fluid flow vector u, modeling the ambient ocean environment, satisfies a Stokes equation, where $P=P(x,t)$ represents the associated pressure, and the buoyancy effect of the sperm and egg on the velocity, mediated through a given gravitational potential $\phi$, is taken into account. We note that the use of the Stokes equation instead of the Navier–Stokes equation is justified by the observation that the fluid flow is relatively slow compared with the movement of the sperm and egg. We further note that the sensitivity tensor $\mathscr {S}(x,\rho ,c)$ may take values that are matrices possibly containing nontrivial off-diagonal entries, which reflects that the chemotactic migration may not necessarily be oriented along the gradient of the chemical signal, but may rather involve rotational flux components (see Xue and Othmer (2009); Xue (2015) for the detailed model derivation).

A two-component variant of (4.1.1) has been used in the mathematical study of coral broadcast spawning. Indeed, in Kiselev and Ryzhik (2012a, b), Kiselev and Ryzhik investigated the important effect of chemotaxis on the coral fertilization process via the Keller–Segel type system of the form

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t+u\cdot \nabla \rho =\varDelta \rho -\chi \nabla \cdot (\rho \nabla c)-\mu \rho ^q, \\&0=\varDelta c+\rho \end{aligned}\right. \end{aligned}$$

(4.1.2)

with a given regular solenoidal fluid flow vector u. This model implicitly assumes that the densities of sperm and egg gametes are identical, and that the Péclet number for the chemical concentration c is small which allows us to ignore the effects of convection on c. The authors showed that, for the Cauchy problem in $\mathbb {R}^2$, the total mass $\int _{\mathbb {R}^2} \rho (x,t)dx$ can become arbitrarily small with increasing $\chi$ in the case $q > 2$ of supercritical reaction, whereas in the critical case $q = 2$, a weaker but related effect within finite time intervals is observed. Recently, Ahn et al. (2017) established the global well-posedness of regular solutions for the variant model of (4.1.2) with $c_t+u\cdot \nabla c=\varDelta c-c+\rho$ instead of $0=\varDelta c+\rho$. They also proved that $\int _{\mathbb {R}^d} \rho (x,t)dx$ $(d=2,3)$ asymptotically approaches a strictly positive constant $C(\chi )$ which tends to 0 as $\chi \rightarrow \infty$.

In Espejo and Suzuki (2015), Espejo and Suzuki studied the three-component variant of (4.1.1)

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t+u\cdot \nabla \rho =\varDelta \rho -\chi \nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)-\mu \rho ^2, \\&c_t+u\cdot \nabla c=\varDelta c-c+\rho , \\&u_t+ \kappa (u\cdot \nabla ) u=\varDelta u-\nabla P+\rho \nabla \phi , \\&\nabla \cdot u=0 \end{aligned}\right. \end{aligned}$$

(4.1.3)

in the modeling of broadcast spawning when the interaction of chemotactic movement of the gametes and the surrounding fluid is not negligible. Here the coefficient $\kappa \in \mathbb {R}$ is related to the strength of nonlinear convection. In particular, when the fluid flow is slow, we can use the Stokes instead of the Navier–Stokes equation, i.e., assume $\kappa = 0$ (see Difrancesco et al. (2010); Lorz (2010)). It should be mentioned that the chemotaxis–fluid model with $c_t+u\cdot \nabla c=\varDelta c-c\rho$ replacing the second equation in (4.1.3) has also been used to describe the behavior of bacteria of the species Bacillus subtilis suspended in sessile water drops (Tuval et al. 2005). From the viewpoint of mathematical analysis, this chemotaxis–fluid system compounds the known difficulties in the study of fluid dynamics with the typical intricacies in the study of chemotaxis systems. It has also been observed that when $\mathscr {S}=\mathscr {S}(x,\rho ,c)$ is a tensor, the corresponding chemotaxis–fluid system loses some energy-like structure, which plays a key role in the analysis of the scalar-valued case. Despite these challenges, some comprehensive results on the global boundedness and large time behavior of solutions are available in the literature (see Cao and Lankeit (2016); Li et al. (2019a); Liu and Wang (2017); Tao and Winkler (2015b); Wang and Xiang (2016); Winkler (2012, 2017b, 2018c, e) for example). It has been shown that when $\mathscr {S}=\mathscr {S}(x,\rho ,c)$ is a tensor fulfilling

$$\begin{aligned} |\mathscr {S}(x,\rho ,c)|\le \frac{C_{\mathscr {S}}}{(1+\rho )^{\alpha }}\quad \text {for some}~\alpha>0~\text {and}~ C_{\mathscr {S}}>0, \end{aligned}$$

(4.1.4)

the three-dimensional system (4.1.3) with $\mu =0$, $\kappa =0$ admits globally bounded weak solutions for $\alpha >1/2$ (Wang and Xiang 2016), which is slightly stronger than the corresponding subcritical assumption $\alpha >1/3$ for the fluid-free system. As for $\alpha \ge 0$, when the suitably regular initial data $(\rho _0,c_0,u_0)$ fulfill a smallness condition, (4.1.3) with $\mu =0$, $\kappa =1$ possesses a global classical solution which decays to $(\bar{\rho }_0,\bar{\rho }_0,0)$ exponentially with $\bar{\rho }_0=\frac{1}{|\varOmega |}\int _{\varOmega } \rho _0(x)dx$ (Yu et al. 2018). Removing the presupposition that the densities of the sperm and egg coincide at each point, Espejo and Suzuki (2017) looked at a simplified version of (4.1.1) in two dimensions, namely

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t+u\cdot \nabla \rho =\varDelta \rho -\chi \nabla \cdot (\rho \nabla c)-\rho m, \\&m_t+u\cdot \nabla m=\varDelta m-\rho m, \\&0=\varDelta c+k_0(m-\displaystyle \frac{1}{|\varOmega |}\int _{\varOmega } m dx) ~\hbox {with}~ \int _{\varOmega } c dx=0, \end{aligned}\right. \end{aligned}$$

(4.1.5)

and showed that $\int _{\varOmega } \rho _0(x)dx\ge \int _{\varOmega } m_0 (x)dx$ implies that m(x, t) vanishes asymptotically, while $\int _{\varOmega } \rho (x,t)dx\rightarrow \frac{1}{|\varOmega |}(\int _{\varOmega } \rho _0(x)dx- \int _{\varOmega } m_0 (x) dx)$ as $t\rightarrow \infty$, provided that $\chi$ is small enough and u is low. In two dimensions, Espejo and Winkler (2018) have recently considered the Navier–Stokes version of (4.1.1)

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t+u\cdot \nabla \rho =\varDelta \rho -\nabla \cdot (\rho \nabla c)-\rho m, \\&m_t+u\cdot \nabla m=\varDelta m-\rho m, \\&c_t+u\cdot \nabla c=\varDelta c-c+m, \\&u_t +\kappa (u\cdot \nabla )=\varDelta u-\nabla P+(\rho +m)\nabla \phi ,\quad \nabla \cdot u=0, \end{aligned}\right. \end{aligned}$$

(4.1.6)

and established the global existence of classical solutions to the associated initial-boundary value problem, which tend toward a spatially homogeneous equilibrium in the large time limit.

In Sect. 4.3, motivated by the above works, we shall consider the properties of solutions to (4.1.1). In particular, we shall show that the corresponding solutions converge to a spatially homogeneous equilibrium exponentially as $t\rightarrow \infty$ as well.

Throughout the rest of this part, we shall assume that

$$\begin{aligned} \left\{ \begin{aligned}&\rho _0\in C^0(\overline{\varOmega }),~\rho _0\ge 0 ~\hbox {and}~ \rho _0\not \equiv 0, \\&m_0\in C^0(\overline{\varOmega }),~m_0\ge 0 ~\hbox {and}~ m_0\not \equiv 0, \\&c_0\in W^{1,\infty }(\varOmega ),~c_0\ge 0 ~\hbox {and}~ c_0\not \equiv 0, \\&u_0\in D(A^{\beta }) ~\hbox {for all}~ \beta \in (\frac{3}{4},1), \end{aligned}\right. \end{aligned}$$

(4.1.7)

where A denotes the realization of the Stokes operator in $L^2(\varOmega )$. Under these assumptions, we shall first establish the existence of global bounded classical solutions to (4.1.1):

Theorem 4.1

Suppose that (4.1.4), (4.1.7) hold with $\alpha >\frac{1}{3}$. Then the system (4.1.1) admits a global classical solution $(\rho ,m,c,u,P)$, which is uniformly bounded in the sense that for any $\beta \in (\frac{3}{4},1)$, there exists $K>0$ such that for all $t\in (0,\infty )$

$$\begin{aligned} \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )} +\Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}\le K. \end{aligned}$$

(4.1.8)

Then, we establish the large time behavior of these solutions as follows:

Theorem 4.2

Under the assumptions of Theorem 4.1, the solutions given by Theorem 4.1 satisfy

$$ \rho (\cdot ,t)\rightarrow \rho _\infty ,~ m(\cdot ,t)\rightarrow m_\infty ,~ c(\cdot ,t)\rightarrow m_\infty ,~ u(\cdot ,t)\rightarrow 0~ \hbox {in}~ L^\infty (\varOmega ) ~\hbox {as}~ t\rightarrow \infty . $$

Furthermore, when $\int _\varOmega \rho _0\ne \int _\varOmega m_0$, there exist $K>0$ and $\delta >0$ such that

$$\begin{aligned} \Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^2(\varOmega )}&\le Ke^{-\delta t},\end{aligned}$$

(4.1.9)

$$\begin{aligned} \Vert m(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}&\le Ke^{-\delta t},\end{aligned}$$

(4.1.10)

$$\begin{aligned} \Vert c(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}&\le Ke^{-\delta t},\end{aligned}$$

(4.1.11)

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}&\le Ke^{-\delta t}, \end{aligned}$$

(4.1.12)

where $\rho _\infty =\frac{1}{|\varOmega |}\left\{ \int _\varOmega \rho _0-\int _\varOmega m_0\right\} _+$, $m_\infty =\frac{1}{|\varOmega |}\left\{ \int _\varOmega m_0-\int _\varOmega \rho _0\right\} _+$.

According to the result for the related fluid–free system, the subcritical restriction $\alpha >\frac{1}{3}$ seems to be necessary for the existence of global bounded solutions. However, for $\alpha \le \frac{1}{3}$, inspired by Cao and Lankeit (2016); Yu et al. (2018), we investigate the existence of global bounded classical solutions and their large time behavior under a smallness assumption imposed on the initial data, which can be stated as follows Li et al. (2019b):

Theorem 4.3

Suppose that (4.1.4) hold with $\alpha =0$ and $\int _{\varOmega }\rho _0\ne \int _{\varOmega }m_0$. Further, let $N=3$ and $p_0\in (\frac{N}{2}, \infty )$, $q_0\in (N,\infty )$ if $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$; and $p_0\in (\frac{2N}{3}, \infty )$, $q_0\in (N,\infty )$ if $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$. Then there exists $\varepsilon >0$ such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.7) as well as

$$ \Vert \rho _0-\rho _\infty \Vert _{L^{p_0}(\varOmega )}\le \varepsilon ,\quad \Vert m_0-m_\infty \Vert _{L^{q_0}(\varOmega )}\le \varepsilon , \quad \Vert \nabla c_0\Vert _{L^{N}(\varOmega )}\le \varepsilon , \quad \Vert u_0\Vert _{L^{N}(\varOmega )}\le \varepsilon , $$

(4.1.1) possesses a global classical solution $(\rho ,m,c,u,P)$. Moreover, for any $\alpha _1$ $\in (0,\min \{\lambda _1, m_\infty +\rho _\infty \})$, $\alpha _2\in (0,\min \{\alpha _1,\lambda _1',1\})$, there exist constants $K_i$, $i=1,2,3,4$, such that for all $t\ge 1$,

$$\begin{aligned} \Vert m(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t},\quad \Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\ \Vert c(\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \quad \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

Here $\lambda '_1$ is the first eigenvalue of A, and $\lambda _1$ is the first nonzero eigenvalue of $-\varDelta$ on $\varOmega$ under the Neumann boundary condition.

Remark 4.1

In Theorem 4.3, we have excluded the case $\int _{\varOmega }\rho _0=\int _{\varOmega }m_0$. Indeed, in this case, some results of Cao and Winkler (2018) suggest that exponential decay of solutions may not hold.

Remark 4.2

It is observed that the similar result to Theorem 4.3 is also valid for the Navier–Stokes version of (4.1.1) upon slight modification of the definition of T in (4.3.60) and (4.3.94).

As mentioned above, compared with the scalar sensitivity $\mathscr {S}$, the system (4.1.1) with rotational tensor loses a favorable quasi-energy structure. For example, we note that the integral

$$ \int _\varOmega \rho ln \rho +a\int _\varOmega |\nabla c|^2+b \int _\varOmega |u|^2 $$

with appropriate positive constants a and b plays a favorable entropy-like functional in deriving the bounds of solution to (4.1.6). However, this will no longer be available in the present situation (see Espejo and Winkler (2018)). To overcome this difficulty, our approach underlying the derivation of Theorem 4.1 will be based on the estimate of the functional

$$\begin{aligned} \Vert \rho (\cdot ,t)\Vert _{L^2(\varOmega )}^2+\Vert u(\cdot ,t)\Vert _{W^{1,2}(\varOmega )}^2+\Vert \nabla c(\cdot ,t)\Vert _{L^2(\varOmega )}^2. \end{aligned}$$

In addition, the proof of the exponential decay results in Theorem 4.2 relies on careful analysis of the functional

$$\begin{aligned} G(t):=\int _\varOmega (\rho -\overline{\rho })^2+a\int _\varOmega (m-\overline{m})^2+b\int _\varOmega (c-\overline{c})^2 +c\int _\varOmega \rho m \end{aligned}$$

with suitable parameters $a,b,c>0$. Indeed, it can be seen that G(t) satisfies the ODE $G'(t)+\delta _1 G(t)\le 0$ for some $\delta _1>0$, and thereby the convergence rate of solutions in $L^2(\varOmega )$ is established. At the same time, in comparison with the chemotaxis–fluid system considered in Cao and Lankeit (2016); Yu et al. (2018), due to

$$\Vert e^{t\varDelta }\omega \Vert _{L^p(\varOmega )}\le c_1\left( 1+t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right) e^{-\lambda _1t}\Vert \omega \Vert _{L^q(\varOmega )} $$

for all $\omega \in L^q(\varOmega )$ with $\int _\varOmega \omega =0$, $-\rho m$ in the first equation of (4.1.1) gives rise to some difficulty in mathematical analysis despite its dissipative feature. Accordingly, it requires a nontrivial application of the mass conservation of $\rho (x,t)-m(x,t)$.

In Sect. 4.4, we are concerned with the asymptotic behavior of classical solutions of the three-dimensional Keller–Segal–Navier–Stokes system

$$\begin{aligned} \left\{ \begin{aligned}&\rho _t+u\cdot \nabla \rho =\varDelta \rho -\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)-\rho m,&(x,t)\in \varOmega \times (0,T), \\&m_t+u\cdot \nabla m=\varDelta m-\rho m,&(x,t)\in \varOmega \times (0,T), \\&c_t+u\cdot \nabla c=\varDelta c-c+m,&(x,t)\in \varOmega \times (0,T), \\&u_t+ (u\cdot \nabla ) u=\varDelta u-\nabla P+(\rho +m)\nabla \phi ,\quad \nabla \cdot u=0,&(x,t)\in \varOmega \times (0,T), \\&(\nabla \rho -\rho \mathscr {S}(x,\rho ,c)\nabla c)\cdot \nu =\nabla m\cdot \nu =\nabla c\cdot \nu =0, u=0,&(x,t)\in \partial \varOmega \times (0,T),\\&\rho (x,0)=\rho _0(x),\quad m(x,0)=m_0(x),\quad c(x,0)=c_0(x),&u(x,0)=u_0(x),\; x\in \varOmega . \end{aligned}\right. \end{aligned}$$

(4.1.13)

In this coral fertilization model, the sperm $\rho$ chemotactically moves toward the higher concentration of the chemical c released by the egg m, while the egg m is merely affected by random diffusion, fluid transport and degradation upon contact with the sperm. We assume that the tensor-valued chemotactic sensitivity $\mathscr {S}=\mathscr {S}(x,\rho ,c)$ satisfies

$$\begin{aligned} |\mathscr {S}(x,\rho ,c)|\le C_{\mathscr {S}} \quad \text {for some}~ C_{\mathscr {S}}>0, \end{aligned}$$

(4.1.14)

and the initial data satisfy

$$\begin{aligned} \left\{ \begin{aligned}&\rho _0\in C^0(\overline{\varOmega }),~\rho _0\ge 0 ~\hbox {and}~ \rho _0\not \equiv 0, \\&m_0\in C^0(\overline{\varOmega }),~m_0\ge 0 ~\hbox {and}~ m_0\not \equiv 0, \\&c_0\in W^{1,\infty }(\varOmega ),~c_0\ge 0 ~\hbox {and}~ c_0\not \equiv 0, \\&u_0\in D(A^{\beta }) ~\hbox {for all}~ \beta \in (\frac{3}{4},1), \end{aligned}\right. \end{aligned}$$

(4.1.15)

where A denotes the realization of the Stokes operator in $L^2(\varOmega )$.

Under these assumptions, our main result can be stated as follows Myowin et al. (2020):

Theorem 4.4

Suppose that (4.1.14) hold and $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$. Let $p_0\in (\frac{3}{2}, 3)$, $q_0\in (3,\frac{3p_0}{3-p_0})$. Then, there exists $\varepsilon >0$ such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.15) as well as

$$\begin{aligned} \Vert \rho _0-\rho _\infty \Vert _{L^{p_0}(\varOmega )}<\varepsilon ,\quad \Vert m_0\Vert _{L^{q_0}(\varOmega )}<\varepsilon , \quad \Vert c_0\Vert _{L^{\infty }(\varOmega )}<\varepsilon , \quad \Vert u_0\Vert _{L^{3}(\varOmega )}<\varepsilon , \end{aligned}$$

(4.1.13) admits a global classical solution $(\rho ,m,c,u,P)$. In particular, for any $\alpha _1\in (0,\min \{\lambda _1,\rho _\infty \})$, $\alpha _2\in (0,\min \{\alpha _1,\lambda _1',1\})$, there exist constants $K_i$, $i=1,2,3,4$, such that for all $t\ge 1$

$$\begin{aligned}&\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t}, \\&\Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\&\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \\&\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

Here, $\lambda _1$ is the first nonzero eigenvalue of $-\varDelta$ on $\varOmega$ under the Neumann boundary condition, $\rho _\infty =\frac{1}{|\varOmega |}(\int _{\varOmega }\rho _0-\int _{\varOmega }m_0)$. and $\lambda '_1$ is the first eigenvalue of A.

As for the case $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$, i.e., $m_\infty =\frac{1}{|\varOmega |}(\int _{\varOmega }m_0-\int _{\varOmega }\rho _0)>0$, we have Myowin et al. (2020)

Theorem 4.5

Assume that (4.1.14) and $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$ hold, and let $p_0\in (2,3)$, $q_0\in (3,\frac{3p_0}{2(3-p_0)})$. Then there exists $\varepsilon >0$ such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.15) as well as

$$\Vert \rho _0\Vert _{L^{p_0}(\varOmega )}\le \varepsilon ,\quad \Vert m_0-m_\infty \Vert _{L^{q_0}(\varOmega )}\le \varepsilon , \quad \Vert \nabla c_0\Vert _{L^{3}(\varOmega )}\le \varepsilon , \quad \Vert u_0\Vert _{L^{3}(\varOmega )}\le \varepsilon , $$

(4.1.13) admits a global classical solution $(\rho ,m,c,u,P)$. Furthermore, for any $\alpha _1\!\in \!(0,\min \{\lambda _1,m_\infty ,1\})$, $\alpha _2\!\in \!(0,\min \{\alpha _1,\lambda _1'\})$, there exist constants $K_i>0$, $i=1,2,3,4$, such that

$$\begin{aligned}&\Vert m(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t}, \\&\Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\&\Vert c(\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \\&\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

Remark 4.3

In our results, we have excluded the case $\int _{\varOmega }\rho _0=\int _{\varOmega }m_0$. Indeed, in light of results of Cao and Winkler (2018); Htwe and Wang (2019), algebraic decay rather than exponential decay of the solutions is expected in this case.

It is noted that the nonlinear convection $(u\cdot \nabla ) u$ in the three-dimensional Navier–Stokes equation may lead to the spontaneous emergence of singularities, resulting in a blow-up with respect to the norm of $L^\infty (\varOmega )$. Hence, we subject the study of classical solutions of (4.1.13) to small initial data. We further note the substantial difference between dimensions two and three, and acknowledge results on global boundedness in two dimensions obtained by Espejo (2018) in the case of scalar-valued sensitivity and by Li (2019) in the case of tensor-valued sensitivity with saturation effect or suitably small initial data.

Section 4.5 is devoted to the large time behavior in a chemotaxis-Stokes system modeling coral fertilization with arbitrarily slow porous medium diffusion. In accordance with the phenomena observed from experiments (Coll et al. 1994, 1995; Miller 1979, 1985), oriented motions may occur to sperms in response to some chemical signal secreted by eggs during the period of coral fertilization. In order to describe this in mathematics, a model appearing as

$$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\nabla \cdot (D(n)\nabla n)-\nabla \cdot (n\nabla c)-nv,\\&c_t+u\cdot \nabla c=\varDelta c-c+v,\\&v_t+u\cdot \nabla v=\varDelta v-vn,\\&u_t=\varDelta u+\nabla P+(n+v)\nabla \varPhi , \end{aligned}\right. \end{aligned}$$

(4.1.16)

was proposed under the assumptions that sperms and eggs enjoy different densities n and v,  respectively, that P and $\varPhi$ separately stand for the liquid pressure and the gravitational potential with

$$\begin{aligned} \varPhi \in W^{2,\infty }(\varOmega ), \end{aligned}$$

(4.1.17)

and that the fluid velocity u is an unknown function (Espejo and Winkler 2018).

For simplified versions of (4.1.16), such as $D\equiv 1$ together with $n\equiv v$ or with a given fluid field u,  related analytical results on global dynamic behaviors of the solution can be found in Espejo and Suzuki (2015, 2017). Whereas for more complex situations, during the past years, a number of analytic approaches have been developed to explore global dynamics in (4.1.16) and the variants thereof.

In particular, under the interaction of linear diffusion, i.e., $D\equiv 1,$ with proper saturation effects of cells, by constructing appropriate weighted functions g and whereafter detecting the evolution of

$$\begin{aligned} \int _{\varOmega }n^p g(c)\end{aligned}$$

(4.1.18)

with any $p>1,$ system (4.1.16) coupled with (Navier–)Stokes-fluid is proved to be globally solvable in the classical sense (Li 2019) or in the weak sense (Zheng 2021). Moreover, arguments based on $L^p$-$L^q$ estimates for Neumann heat semigroup further show exponential decay features of the corresponding classical solutions under suitable smallness assumptions on initial data (Htwe et al. 2020; Li et al. 2019b).

As a more frequently used method, energy-based arguments, which start from constructions of proper energy functionals, play a crucial role in the whole study of systems related to (4.1.16). More precisely, as shown in Espejo and Winkler (2018), an analysis of a suitably established entropy-like functional

$$\begin{aligned} \int _{\varOmega }n\ln n+k_1\int _{\varOmega }|\nabla c|^2+k_2\int _{\varOmega }|u|^2 \end{aligned}$$

(4.1.19)

with $k_1>0$ and $k_2>0$ underlies the derivation of global boundedness and stabilization of the unique classical solution to the Navier–Stokes version of system (4.1.16) with $D\equiv 1$ in spatially two-dimensional setting. In cases when saturation influence of cells is accounted for in the cross-diffusion term of n-equation, the construction of a similar but different functional as compared to (4.1.19) is also viewed as the fundament in deriving global solvability of system (4.1.16) with $D\equiv 1,$ both in the Stokes-fluid context (Li et al. 2019b) and in the Navier–Stokes-fluid setting (Liu et al. 2020). Apart from that, when cell mobility depends on gradients of some unknown quantity, such as p-Laplacian cell diffusion, the pursuance of global solvability involves an analysis of a functional with more complex structure (Liu 2020).

Actually, whether by establishing weighted estimates as (4.1.18) or by constructing energy functionals of different types, the core of the analysis is to derive a uniform $L^p$ bound of component n for any $p>1.$ Taking a recent work (Liu 2020) as an example, in the presence of a porous medium type diffusion, namely D in (4.1.16) is chosen to generalize the prototypical case

$$\begin{aligned} D(s)=s^{m-1},\quad s>0 \end{aligned}$$

(4.1.20)

with some $m>1,$ the condition

$$\begin{aligned} m>\frac{37}{33} \end{aligned}$$

(4.1.21)

therein reflects an explicitly quantitative requirement for the strength of nonlinear diffusion in the derivation of temporally independent $L^p$ estimates for n. However, since complementary results on possibly emerging explosion phenomena are rather barren, it is still unknown that corresponding uniform $L^p$ bounds could be achieved for smaller values of m or even for the optimal restriction $m\ge 1.$

In the present work, we attempt to make use of a different method, by which conditional estimates for u and c subject to some uniform $L^p$ norms of n are established, to explore how far the porous medium type diffusion of sperms can prevent the occurrence of singularity formation phenomena.

For precisely formulating our main results, let us close the considered problem involving system (4.1.16) with the following initial-boundary conditions

$$\begin{aligned} n(x,0)=n_0(x),c(x,0)=c_0(x),v(x,0)=v_0(x)~~\text {and}~~u(x,0)=u_0(x),\quad x\in \varOmega \end{aligned}$$

(4.1.22)

as well as

$$\begin{aligned} D(n)\frac{\partial n}{\partial \nu }=\frac{\partial c}{\partial \nu }=\frac{\partial v}{\partial \nu }=0~~\text {and}~~u=0,\quad x\in \partial \varOmega ,~t>0, \end{aligned}$$

(4.1.23)

where $\varOmega \subset \mathbb {R}^3$ is a bounded domain with smooth boundary, where the function D fulfills

$$\begin{aligned} D\in C^{\mu }_{loc}([0,\infty ))\bigcap C^2_{loc}((0,\infty ))~~\text {and}~~D(s)\ge C_Ds^{m-1}~~\text {for any}~~s\ge 0 \end{aligned}$$

(4.1.24)

with certain $\mu \in (0,1),C_D>0$ and $m\ge 1,$ and where the initial data satisfies

$$\begin{aligned} \left\{ \begin{aligned}&n_0 \in C^{\nu }({\bar{\varOmega }})~~\text {for some}~~\nu >0~~\text {with}~~n_0\ge 0~~\text {in}~~\varOmega ~~\text {and}~~n_0\not \equiv 0,\\&c_0 \in W^{1,\infty }(\varOmega )~~\text {with}~~c_0\ge 0~~\text {in}~~\varOmega ,\\&v_0 \in W^{1,\infty }(\varOmega )~~\text {with}~~v_0\ge 0~~\text {in}~~\varOmega ,~~\text {and}\\&u_0 \in D(A^{\alpha })~~\text {for certain}~~\alpha \in \left( \frac{3}{4},1\right) \end{aligned}\right. \end{aligned}$$

(4.1.25)

with A representing the realization of the Stokes operator with its domain defined as $D(A):=W^{2,2}(\varOmega ;\mathbb {R}^3)\bigcap W^{1,2}_0(\varOmega ;\mathbb {R}^3)\bigcap L^2_{\sigma }(\varOmega )$ with $L^2_{\sigma }(\varOmega ):=\{\omega \in L^2(\varOmega ;\mathbb {R}^3)|\nabla \cdot \omega =0\}$ (Sohr 2001).

Within this framework, our main results can be read as follows (Wang and Liu 2022).

Theorem 4.6

Assume that $\varOmega \subset \mathbb {R}^3$ is a bounded domain with smooth boundary. Let (4.1.17) be satisfied, and let (4.1.24) hold with

$$\begin{aligned} m>1.\end{aligned}$$

(4.1.26)

Then for each $(n_0,c_0,v_0,u_0)$ complying with (4.1.25), there exist functions n, c, v and u fulfilling

$$\begin{aligned} \left\{ \begin{aligned}&n\in L^{\infty }(\varOmega \times (0,\infty ))\cap C^0([0,\infty );(W^{2,2}_0(\varOmega ))^{*}),\\&c\in \cap _{r>3} L^{\infty }((0,\infty );W^{1,r}(\varOmega ))\cap C^0({\bar{\varOmega }}\times [0,\infty ))\cap C^{1,0}({\bar{\varOmega }}\times (0,\infty )),\\&v\in \cap _{r>3} L^{\infty }((0,\infty );W^{1,r}(\varOmega ))\cap C^0({\bar{\varOmega }}\times [0,\infty ))\cap C^{1,0}({\bar{\varOmega }}\times (0,\infty )),\\&u\in L^{\infty }(\varOmega \times (0,\infty );\mathbb {R}^3)\cap L^2_{loc}([0,\infty );W^{1,2}_{0}(\varOmega ;\mathbb {R}^3)\cap L^2_{\sigma }(\varOmega ))\cap C^0({\bar{\varOmega }}\times [0,\infty );\mathbb {R}^3), \end{aligned}\right. \end{aligned}$$

(4.1.27)

such that $n\ge 0,c\ge 0$ and $v\ge 0,$ and that along with certain $P\in C^{0}\left( \varOmega \times (0,\infty )\right)$ the quintuple (n, c, v, u, P) becomes a global weak solution of the problem (4.1.16), (4.1.22) and (4.1.23) in the sense of Definition 4.1 below, and has the stabilization features that

$$\begin{aligned} \Vert n(\cdot ,t)-n_{\infty }\Vert _{L^{p}(\varOmega )}+\Vert c(\cdot ,t)-v_{\infty }\Vert _{W^{1,\infty }(\varOmega )} +\Vert v(\cdot ,t)-v_{\infty }\Vert _{W^{1,\infty }(\varOmega )}+\Vert u(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\rightarrow 0 \end{aligned}$$

(4.1.28)

for any $p\ge 1$ as $t\rightarrow \infty$ with

$$n_{\infty }:=\frac{1}{|\varOmega |}\left\{ \int _{\varOmega }n_0-\int _{\varOmega }v_0\right\} _{+}\quad \hbox {and}\; v_{\infty }:=\frac{1}{|\varOmega |}\left\{ \int _{\varOmega }v_0-\int _{\varOmega }n_0\right\} _{+}.$$

From (4.1.26), which shows the values that m could be taken herein for successfully establishing temporally independent $L^p$ bounds of n,  one can see that an apparent relaxation is realized in comparison to the previously derived range of m,  i.e., (4.1.21). In fact, for introduced approximated problems of (4.1.16), (4.1.22) and (4.1.23), which is verified to be locally solvable with an extensible blow-up criterion, the hypothesis (4.1.26) allows for an application of a standard testing procedure to derive the uniform $L^p$ estimates of $(n_{\varepsilon })_{\varepsilon \in (0,1)}$ with the aids of conditionally uniform $L^{\infty }$ estimates of $(\nabla c_{\varepsilon })_{\varepsilon \in (0,1)}$ which are established by utilizing $L^p$-$L^q$ estimates for fractional powers of a sectorial operator on the basis of basic estimates implied in the regularized problems and of some well-established conditional estimates of $(u_{\varepsilon })_{\varepsilon \in (0,1)}$ (see Sects. 4.3–4.4). The derivation of (4.1.28) is essentially based on the dissipative effect of the considered consumption process, as shown in Espejo and Winkler (2018) for two-dimensional Navier–Stokes version of (4.1.16) with $m=1,$ or in Winkler (2015b, 2018c) and Winkler (2014b, 2017b, 2021a) for simplified oxygen-consumption type chemotaxis-fluid models with $m>1$ and $m=1,$ respectively. More precisely, the absorptive contribution $-nv$ to the third equation in (4.1.16) implies time-independently uniform bounds of spatio-temporal integrals for nv and for the square of the gradients of both v and c, which underlies the achievement of the convergence of n, c and v in (4.1.28). Thanks to the convergence of n and v in (4.1.28), the large time behavior of u can be detected by means of a combination of variation-of-constants formula with regularity properties of analytic semigroup.

4.2 Preliminaries

In this subsection, we provide some preliminary results that will be used in the subsequent sections.

Next we introduce the Stokes operator and recall estimates for the corresponding semigroup. With $L_\sigma ^p(\varOmega ):=\{\varphi \in L^p(\varOmega )|\nabla \cdot \varphi =0\}$ and $\mathscr {P}$ representing the Helmholtz projection of $L^p(\varOmega )$ onto $L_\sigma ^p(\varOmega )$, the Stokes operator on $L_\sigma ^p(\varOmega )$ is defined as $A_p=-\mathscr {P}\varDelta$ with domain $D(A_p):=W^{2,p}(\varOmega )\cap W^{2,p}_0(\varOmega )\cap L_\sigma ^p(\varOmega )$. Since $A_{p_1}$ and $A_{p_2}$ coincide on the intersection of their domains for $p_1$, $p_2\in (1,\infty )$, we will drop the index in the following.

Lemma 4.1

(Lemma 4.2 of Cao and Lankeit (2016)) The Stokes operator A generates the analytic semigroup $(e^{-tA})_{t>0}$ in $L_\sigma ^r(\varOmega )$. Its spectrum satisfies $\lambda _1'=\hbox {inf}~ \hbox {Re}\sigma (A)>0$ and we fix $\mu \in (0,\lambda _1')$. For any such $\mu$, we have

(i) For any $p\in (1,\infty )$ and $\gamma \ge 0$, there is $c_5(p,\gamma )>0$ such that for all $\phi \in L^p_\sigma (\varOmega )$,

$$\Vert A^\gamma e^{-tA}\phi \Vert _{L^p(\varOmega )}\le c_5(p,\gamma ) t^{-\gamma }e^{-\mu t}\Vert \phi \Vert _{L^p(\varOmega )};$$

(ii) For any p, q with $1<p\le q<\infty$, there is $c_6(p,q)>0$ such that for all $\phi \in L^p_\sigma (\varOmega )$,

$$\Vert e^{-tA}\phi \Vert _{L^q(\varOmega )}\le c_6(p,q) t^{-\frac{N}{2}\left( \frac{1}{p}-\frac{1}{q}\right) }e^{-\mu t}\Vert \phi \Vert _{L^p(\varOmega )};$$

(iii) For any p, q with $1<p\le q<\infty$, there is $c_7(p,q)>0$ such that for all $\phi \in L^p_\sigma (\varOmega )$,

$$\Vert \nabla e^{-tA}\phi \Vert _{L^q(\varOmega )}\le c_7(p,q) t^{-\frac{1}{2}-\frac{N}{2}\left( \frac{1}{p}-\frac{1}{q}\right) }e^{-\mu t}\Vert \phi \Vert _{L^p(\varOmega )};$$

(iv) If $\gamma \ge 0$ and $1<p<q<\infty$ satisfy $2\gamma -\frac{N}{q}\ge 1-\frac{N}{p}$, there is $c_8(\gamma ,p,q)>0$ such that for all $\phi \in D(A_q^\gamma )$,

$$\Vert \phi \Vert _{W^{1,p}(\varOmega )}\le c_8(\gamma ,p,q)\Vert A^\gamma \phi \Vert _{L^q(\varOmega )}.$$

Lemma 4.2

(Theorem 1 and Theorem 2 of Fujiwara and Morimoto (1977)) The Helmholtz projection $\mathscr {P}$ defines a bounded linear operator $\mathscr {P}$: $L^p(\varOmega )\rightarrow L^p_\sigma (\varOmega )$; in particular, for any $p\in (1,\infty )$, there exists $c_9(p)>0$ such that $\Vert \mathscr {P}\omega \Vert _{L^p(\varOmega )}\le c_9(p)\Vert \omega \Vert _{L^p(\varOmega )}$ for every $\omega \in L^p(\omega )$.

The following elementary lemma provides some useful information on both the short time and the large time behavior of certain integrals, which is used in the proof of Theorem 4.3.

Lemma 4.3

(Lemma 1.2 of Winkler (2010)) Let $\alpha <1$, $\beta <1$, and $\gamma$, $\delta$ be positive constants such that $\gamma \ne \delta$. Then there exists $c_{10}(\alpha ,\beta ,\gamma ,\delta ) > 0$ such that

$$ \begin{aligned}&\int _0^t(1+s^{-\alpha })(1+(t-s)^{-\beta })e^{-\gamma s}e^{-\delta (t-s)}ds \\ \le&c_{10}(\alpha ,\beta ,\gamma ,\delta ) \left( 1+t^{\min \{0,1-\alpha -\beta \}}\right) e^{-\min \{\gamma ,\delta \}t}. \end{aligned} $$

4.3 Global Boundedness and Decay Property of Solutions to a 3D Coral Fertilization Model

4.3.1 A Convenient Extensibility Criterion

At the beginning, we recall the result of the local existence of classical solutions, which can be proved by a straightforward adaptation of a well-known fixed point argument (see Winkler (2012) for example).

Lemma 4.4

Suppose that (4.1.4), (4.1.7) and

$$\begin{aligned} \mathscr {S}(x,\rho ,c)=0, ~~(x,\rho ,c)\in \partial \varOmega \times [0,\infty )\times [0,\infty ) \end{aligned}$$

(4.3.1)

hold. Then there exist $T_{max}\in (0,\infty ]$ and a classical solution $(\rho ,m,c,u,P)$ of (4.1.1) on $(0,T_{max})$. Moreover, $\rho ,m,c$ are nonnegative in $\varOmega \times (0,T_{max})$, and if $T_{max}<\infty$, then for $\beta \in (\frac{3}{4},1)$,

$$\lim _{t\rightarrow T_{max}}\left( \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}\right) =\infty .$$

This solution is unique, up to addition of constants to P.

The following elementary properties of the solutions in Lemma 4.4 are immediate consequences of the integration of the first and second equations in (4.1.1), as well as an application of the maximum principle to the second and third equations.

Lemma 4.5

Suppose that (4.1.4), (4.1.7) and (4.3.1) hold. Then for all $t\in (0,T_{max})$, the solution of (4.1.1) from Lemma 4.4 satisfies

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^1(\varOmega )}\le \Vert \rho _0\Vert _{L^1(\varOmega )},\quad \Vert m(\cdot ,t)\Vert _{L^1(\varOmega )}\le \Vert m_0\Vert _{L^1(\varOmega )}, \end{aligned}$$

(4.3.2)

$$\begin{aligned}&\int _0^t\Vert \rho (\cdot ,s)m(\cdot ,s)\Vert _{L^1(\varOmega )}ds\le \min \{\Vert \rho _0\Vert _{L^1(\varOmega )},\Vert m_0\Vert _{L^1(\varOmega )}\}, \end{aligned}$$

(4.3.3)

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^1(\varOmega )}-\Vert m(\cdot ,t)\Vert _{L^1(\varOmega )}=\Vert \rho _0\Vert _{L^1(\varOmega )}-\Vert m_0\Vert _{L^1(\varOmega )}, \end{aligned}$$

(4.3.4)

$$\begin{aligned}&\Vert m(\cdot ,t)\Vert _{L^2(\varOmega )}^2+2\int _0^t\Vert \nabla m(\cdot ,s)\Vert _{L^2(\varOmega )}^2ds\le \Vert m_0\Vert _{L^2(\varOmega )}^2, \end{aligned}$$

(4.3.5)

$$\begin{aligned}&\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \Vert m_0\Vert _{L^\infty (\varOmega )}, \end{aligned}$$

(4.3.6)

$$\begin{aligned}&\Vert c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \max \{\Vert m_0\Vert _{L^\infty (\varOmega )},\Vert c_0\Vert _{L^\infty (\varOmega )}\}. \end{aligned}$$

(4.3.7)

4.3.2 Global Boundedness and Decay for $\mathscr {S}=0$ on $\partial \varOmega$

In this subsection, we shall consider the case in which besides (4.1.4), the sensitivity satisfies $\mathscr {S}=0$ on $\partial \varOmega$. Under this hypothesis, the boundary condition for $\rho$ in (4.1.1) actually reduces to the homogeneous Neumann condition $\nabla \rho \cdot \nu =0$.

1. Global boundedness for $\mathscr {S}=0$ on $\partial \varOmega$

Lemma 4.6

Suppose that (4.1.4), (4.1.7), (4.3.1) hold with $\alpha >\frac{1}{3}$. Then for any $\varepsilon >0$, there exists $K(\varepsilon )>0$ such that, for all $t\in (0,T_{max})$, the solution of (4.1.1) satisfies

$$\begin{aligned}&\frac{d}{dt}\Vert \rho (\cdot ,t)\Vert _{L^2(\varOmega )}^2+\frac{1}{2}\Vert \nabla \rho (\cdot ,t)\Vert _{L^2(\varOmega )}^2\le \varepsilon \Vert \varDelta c(\cdot ,t)\Vert _{L^2(\varOmega )}^2+K(\varepsilon ). \end{aligned}$$

(4.3.8)

Proof

Multiplying the first equation of (4.1.1) by $\rho$, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\varOmega \rho ^2+\int _\varOmega |\nabla \rho |^2\nonumber \\ =&\int _\varOmega \rho \mathscr {S}(x,\rho ,c)\nabla \rho \nabla c-\int _\varOmega \rho ^2m \\ \le&\frac{1}{2}\int _\varOmega |\nabla \rho |^2+\frac{C_S^2}{2}\int _\varOmega \frac{\rho ^2}{(1+\rho )^{2\alpha }}|\nabla c|^2.\nonumber \end{aligned}$$

(4.3.9)

Now we estimate the term $\frac{C_S^2}{2}\int _\varOmega \frac{\rho ^2}{(1+\rho )^{2\alpha }}|\nabla c|^2$ on the right-hand side of (4.3.9). In fact, if $\alpha \ge \frac{3}{4}$,

$$\begin{aligned} \frac{C_S^2}{2}\int _\varOmega \frac{\rho ^2}{(1+\rho )^{2\alpha }}|\nabla c|^2\le \varepsilon \int _\varOmega |\nabla c|^4+K(\varepsilon ), \end{aligned}$$

(4.3.10)

while for $\alpha \in \left( \frac{1}{3},\frac{3}{4}\right)$,

$$\begin{aligned}&\frac{C_S^2}{2}\int _\varOmega \frac{\rho ^2}{(1+\rho )^{2\alpha }}|\nabla c|^2\le \frac{C_S^2}{2}\int _\varOmega \rho ^{2-2\alpha }|\nabla c|^2 \nonumber \\ \le&\frac{C_S^4}{16\varepsilon }\int _\varOmega \rho ^{4-4\alpha }+\varepsilon \int _\varOmega |\nabla c|^4. \end{aligned}$$

(4.3.11)

On the other hand, by Lemma 4.5 and the Gagliardo–Nirenberg inequality, we get

$$\begin{aligned}&\int _\varOmega |\nabla c|^4 \le C_{GN}\left\{ \Vert \varDelta c\Vert _{L^2(\varOmega )}^2\Vert c\Vert _{L^\infty (\varOmega )}^2+\Vert c\Vert _{L^\infty (\varOmega )}^4\right\} \\ \le&C_{GN}'(\Vert \varDelta c\Vert _{L^2(\varOmega )}^2+1) \nonumber \end{aligned}$$

(4.3.12)

and

$$\begin{aligned} \int _\varOmega |\rho |^{4-4\alpha }=\Vert \rho \Vert _{L^{4-4\alpha }(\varOmega )}^{4-4\alpha }&\le C_{GN}\left\{ \Vert \nabla \rho \Vert _{L^2(\varOmega )}^{(4-4\alpha )\lambda _2}\Vert \rho \Vert _{L^1(\varOmega )}^{(4-4\alpha )(1-\lambda _2)} +\Vert \rho \Vert _{L^1(\varOmega )}^{4-4\alpha }\right\} \end{aligned}$$

with $\lambda _2=\frac{6(3-4\alpha )}{5(4-4\alpha )}$. Due to $\alpha \in \left( \frac{1}{3},\frac{3}{4}\right)$, we have $(4-4\alpha )\lambda _2<2$ and thus

$$\begin{aligned} \frac{C_S^4}{16\varepsilon }\int _\varOmega |\rho |^{4-4\alpha }\le \frac{1}{4}\int _\varOmega |\nabla \rho |^2+K_1 \end{aligned}$$

(4.3.13)

by the Young inequality. Combining (4.3.9)–(4.3.13), we readily have (4.3.8).

Lemma 4.7

Under the assumptions of Lemma 4.6, there exists a positive constant $C = C(m_0,c_0)$ such that for all $t\in (0,T_{max})$, the solution of (4.1.1) satisfies

$$\begin{aligned}&\frac{d}{dt}\Vert \nabla c(\cdot ,t)\Vert _{L^2(\varOmega )}^2+2\Vert \nabla c(\cdot ,t)\Vert _{L^2(\varOmega )}^2+\Vert \varDelta c(\cdot ,t)\Vert _{L^2(\varOmega )}^2 \nonumber \\ \le&K(\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}^2+1). \end{aligned}$$

(4.3.14)

Proof

Multiplying the c-equation of (4.1.1) by $-\varDelta c$ and by the Young inequality, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\varOmega |\nabla c|^2+\int _\varOmega |\varDelta c|^2+\int _\varOmega |\nabla c|^2\nonumber \\ \le&-\int _\varOmega m\varDelta c+\int _\varOmega (u\cdot \nabla c)\varDelta c \\ =&-\int _\varOmega m\varDelta c-\int _\varOmega \nabla c\cdot (\nabla u\cdot \nabla c)- \int _\varOmega \nabla c\cdot (D^2c\cdot u) \nonumber \\ =&-\int _\varOmega m\varDelta c-\int _\varOmega \nabla c\cdot (\nabla u\cdot \nabla c) \nonumber \\ \le&\int _\varOmega |m|^2+\frac{1}{4}\int _\varOmega |\varDelta c|^2+(\int _\varOmega |\nabla c|^4)^{\frac{1}{2}}(\int _\varOmega |\nabla u|^2)^{\frac{1}{2}}\nonumber \\ \le&\Vert m\Vert _{L^2(\varOmega )}^2+\frac{1}{4}\Vert \varDelta c\Vert _{L^2(\varOmega )}^2+\frac{1}{2\varepsilon }\Vert \nabla u\Vert _{L^2(\varOmega )}^2+ \frac{\varepsilon }{2}\Vert \nabla c\Vert _{L^4(\varOmega )}^4,\nonumber \end{aligned}$$

(4.3.15)

where the fact that u is solenoidal and vanishes on $\partial \varOmega$ is used to ensure $\int _\varOmega \nabla c\cdot (D^2c\cdot u)=0$.

By (4.3.12) and taking $\varepsilon =\frac{1}{2C_{GN}'}$ in the above inequality, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _\varOmega |\nabla c|^2+\frac{1}{2}\int _\varOmega |\varDelta c|^2+\int _\varOmega |\nabla c|^2\le \Vert m\Vert _{L^2(\varOmega )}^2+C_{GN}'\Vert \nabla u\Vert _{L^2(\varOmega )}^2+\frac{1}{4}, \end{aligned}$$

which along with (4.3.5) readily ensures the validity of (4.3.14).

Lemma 4.8

Under the assumptions of Lemma 4.6, the solution of (4.1.1) satisfies

$$\begin{aligned} \frac{d}{dt}\Vert u(\cdot ,t)\Vert _{L^2(\varOmega )}^2+\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}^2\le&K\left( \Vert \rho (\cdot ,t)\Vert _{L^2(\varOmega )}^2+1\right) , \end{aligned}$$

(4.3.16)

$$\begin{aligned} \frac{d}{dt}\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}^2+\Vert A u(\cdot ,t)\Vert _{L^2(\varOmega )}^2\le&K\left( \Vert \rho (\cdot ,t)\Vert _{L^2(\varOmega )}^2+1\right) \end{aligned}$$

(4.3.17)

for all $t\in (0,T_{max})$ for a positive constant K.

Proof

Testing the u-equation in (4.1.1) by u, using the Hölder inequality and Poincaré inequality, we can get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _\varOmega |u|^2+\int _\varOmega |\nabla u|^2&=\int _\varOmega (\rho +m)\nabla \phi \cdot u \\&\le \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\Vert \rho +m\Vert _{L^2(\varOmega )}\Vert u\Vert _{L^2(\varOmega )} \\&\le \frac{1}{2}\Vert \nabla u\Vert _{L^2(\varOmega )}^2+K_1(\Vert \rho \Vert _{L^2(\varOmega )}^2+\Vert m\Vert _{L^2(\varOmega )}^2), \end{aligned}$$

which together with (4.3.5) yields (4.3.16). Applying the Helmholtz projection $\mathscr {P}$ to the fourth equation in (4.1.1), testing the resulting identity by Au and using the Young inequality, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _{\varOmega }|A^{\frac{1}{2}} u|^2+\int _\varOmega |A u|^2&=-\int _\varOmega \mathscr {P}[(\rho +m)\nabla \phi ] \cdot A u \\&\le \frac{1}{2}\int _\varOmega |A u|^2+K_2(\int _\varOmega \rho ^2+\int _\varOmega m^2), \end{aligned}$$

which yields (4.3.17), due to (4.3.5) and the fact that $\int _{\varOmega }|\nabla u|^2=\int _\varOmega |A^{\frac{1}{2}} u|^2$.

Lemma 4.9

Under the assumptions of Lemma 4.6, one can find $C>0$ such that for all $t\in (0,T_{max})$, the solution of (4.1.1) satisfies

$$\begin{aligned} \Vert \rho (\cdot ,t)\Vert _{L^2(\varOmega )}^2+\Vert \nabla c(\cdot ,t)\Vert _{L^2(\varOmega )}^2+\Vert u(\cdot ,t)\Vert _{W^{1,2}(\varOmega )}^2\le K. \end{aligned}$$

Proof

By the Gagliardo–Nirenberg inequality

$$\begin{aligned} \Vert \rho \Vert _{L^2(\varOmega )}\le C_{GN}\left( \Vert \nabla \rho \Vert _{L^2(\varOmega )}^{\frac{3}{5}}\Vert \rho \Vert _{L^1(\varOmega )}^{\frac{2}{5}}+\Vert \rho \Vert _{L^1(\varOmega )}\right) \end{aligned}$$

and (4.3.8), for any $\varepsilon >0$, there exists $K(\varepsilon )>0$ such that

$$\begin{aligned}&\frac{d}{dt}\Vert \rho \Vert _{L^2(\varOmega )}^2+\Vert \rho \Vert _{L^2(\varOmega )}^2+\frac{1}{4}\Vert \nabla \rho \Vert _{L^2(\varOmega )}^2\le \varepsilon \Vert \varDelta c\Vert _{L^2(\varOmega )}^2+K_1(\varepsilon ). \end{aligned}$$

(4.3.18)

Adding (4.3.16) and (4.3.17), and by the Poincaré inequality, one can find constants $K_i>0$, $i=2,3,4$, such that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}(\Vert u\Vert _{L^2(\varOmega )}^2+\Vert \nabla u\Vert _{L^2(\varOmega )}^2)+K_2(\Vert u\Vert _{L^2(\varOmega )}^2+\Vert \nabla u\Vert _{L^2(\varOmega )}^2) \\ \le&K_3\left( \Vert \rho \Vert _{L^2(\varOmega )}^2+1\right) \\ \le&\displaystyle \frac{1}{8}\Vert \nabla \rho \Vert _{L^2(\varOmega )}^2+K_4. \end{aligned} \end{aligned}$$

(4.3.19)

Recalling (4.3.14), we get

$$\begin{aligned} \frac{d}{dt}\Vert \nabla c\Vert _{L^2(\varOmega )}^2+2\Vert \nabla c\Vert _{L^2(\varOmega )}^2+\Vert \varDelta c\Vert _{L^2(\varOmega )}^2\le K_5\left( \Vert \nabla u\Vert _{L^2(\varOmega )}^2+1\right) . \end{aligned}$$

(4.3.20)

Now combining the above inequalities and choosing $\varepsilon =\frac{K_2}{2K_5}$, one can see that there exists some constant $K_6>0$ such that

$$\begin{aligned} Y(t):=\Vert \rho (\cdot ,t)\Vert _{L^2(\varOmega )}^2+\Vert u(\cdot ,t)\Vert _{W^{1,2}(\varOmega )}^2 +\varepsilon \Vert \nabla c(\cdot ,t)\Vert _{L^2(\varOmega )}^2 \end{aligned}$$

satisfies $Y'(t)+\delta Y(t)\le K_6,$ where $\delta =\min \{1,\frac{K_2}{2}\}$. Hence, by an ODE comparison argument, we obtain $Y(t)\le K_7$ for some constant $K_7>0$ and thereby complete the proof.

With all of the above estimates at hand, we can now establish the global existence result in the case $\mathscr {S}=0$ on $\partial \varOmega$.

Proof of Theorem 4.1 in the case $\mathscr {S}=0$ on $\partial \varOmega$.   To establish the existence of globally bounded classical solution, by the extensibility criterion in Lemma 4.4, we only need to show that

$$\begin{aligned} \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )} +\Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}\le K_1 \end{aligned}$$

(4.3.21)

for all $t\in (0,T_{max})$ with some positive constant $K_1$ independent of $T_{max}$. To this end, by the estimate of Stokes operator (Corollary 3.4 of Winkler (2015b)), we first get

$$\begin{aligned} \Vert u\Vert _{L^\infty (\varOmega )}\le K_2\Vert u\Vert _{W^{1,5}(\varOmega )}\le K_3 \end{aligned}$$

(4.3.22)

with positive constant $K_3>0$ independent of $T_{max}$, due to $\Vert \rho \Vert _{L^2(\varOmega )}\le K_4$ and $\Vert m\Vert _{L^\infty (\varOmega )}\le K_4$ from Lemma 4.9 and Lemma 4.5, respectively.

By Lemma 1.1, Lemma 4.9 and the Young inequality, we have

$$\begin{aligned} \sup _{t\in (0,T_{max})}\Vert \nabla c\Vert _{L^\infty (\varOmega )}&\le K_5(1+\sup _{t\in (0,T_{max})}\Vert m-u\cdot \nabla c\Vert _{L^4(\varOmega )}) \\&\le K_5(1+\sup _{t\in (0,T_{max})}(\Vert m\Vert _{L^4(\varOmega )}+\Vert u\Vert _{L^6(\varOmega )}\Vert \nabla c\Vert _{L^{12}(\varOmega )})) \\&\le K_5(1+\sup _{t\in (0,T_{max})}(\Vert m\Vert _{L^4(\varOmega )}+\Vert u\Vert _{L^6(\varOmega )}\Vert \nabla c\Vert _{L^2(\varOmega )}^{\frac{1}{6}}\Vert \nabla c\Vert _{L^\infty (\varOmega )}^{\frac{5}{6}})) \\&\le K_6(1+\sup _{t\in (0,T_{max})}\Vert \nabla c\Vert _{L^\infty (\varOmega )}^{\frac{5}{6}}), \end{aligned}$$

which implies that $\displaystyle \sup _{t\in (0,T_{max})}\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_7$. Along with (4.3.7) this implies $\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le K_8$. Furthermore, applying the variation-of-constants formula to the $\rho -$equation in (4.1.1), the maximum principle, Lemma 1.1(iv) and Lemma 4.9, we get

$$\begin{aligned} \Vert \rho \Vert _{L^{\infty }(\varOmega )} \le&\Vert e^{t\varDelta }\rho _0\Vert _{L^{\infty }(\varOmega )} +\int ^t_0\Vert e^{(t-s)\varDelta }\nabla \cdot ( \rho \mathscr {S}\nabla c+\rho u)\Vert _{L^{\infty }(\varOmega )}ds\\ \le&\Vert \rho _0\Vert _{L^{\infty }(\varOmega )} + c_4\int ^t_0 (1+ (t-s)^{-\frac{7}{8}}) e^{-\lambda _1(t-s)} \Vert \rho \mathscr {S} \nabla c+\rho u\Vert _{L^4(\varOmega )}ds\\ \le&\Vert \rho _0\Vert _{L^{\infty }(\varOmega )} + K_9\int ^t_0 (1+ (t-s)^{-\frac{7}{8}}) e^{-\lambda _1(t-s)} \Vert \rho \Vert _{L^4(\varOmega )}ds\\ \le&\Vert \rho _0\Vert _{L^{\infty }(\varOmega )} + K_9\int ^t_0 (1+ (t-s)^{-\frac{7}{8}}) e^{-\lambda _1(t-s)} \Vert \rho \Vert ^{\frac{1}{2}}_{L^\infty (\varOmega )}\Vert \rho \Vert ^{\frac{1}{2}}_{L^2(\varOmega )}ds\\ \le&\Vert \rho _0\Vert _{L^{\infty }(\varOmega )} +K_{10}\sup _{s\in (0,T_{max})}\Vert \rho \Vert ^{\frac{1}{2}}_{L^\infty (\varOmega )} \end{aligned}$$

with $K_{10}=K_9\displaystyle \sup _{t\in (0,T_{max})}\Vert \rho \Vert ^{\frac{1}{2}}_{L^2(\varOmega )} \int ^\infty _0 (1+ s^{-\frac{7}{8}}) e^{-\lambda _1s} ds$, where we have used $\nabla \cdot u=0$. Taking supremum on the left-hand side of the above inequality over $(0,T_{max})$, we obtain

$$ \sup _{t\in (0,T_{max})}\Vert \rho \Vert _{L^{\infty }(\varOmega )}\le \Vert \rho _0\Vert _{L^{\infty }(\varOmega )} +K_{10}\sup _{t\in (0,T_{max})}\Vert \rho \Vert ^{\frac{1}{2}}_{L^\infty (\varOmega )}, $$

and thereby $\displaystyle \sup _{t\in (0,T_{max})}\Vert \rho \Vert _{L^{\infty }(\varOmega )}\le K_{11}$ by the Young inequality. Finally, by a straightforward argument (see [Espejo and Winkler (2018), Lemma 3.1] or [Tuval et al. (2005), p. 340]), one can find $K_{12}>0$ such that $\displaystyle \sup _{t\in (0,T_{max})}\Vert A^\beta u\Vert _{L^2(\varOmega )}\le K_{12}$. The boundedness estimate (4.3.21) is now a direct consequence of the above inequalities and this completes the proof.

2. Large time behavior for $\mathscr {S}=0$ on $\partial \varOmega$

This subsection is devoted to showing the large time behavior of global solutions to (4.1.1) obtained in the above subsection. In order to derive the convergence properties of the solution with respect to the norm in $L^2(\varOmega )$, we shall make use of the following lemma. In the sequel, we denote $\overline{f}=\frac{1}{|\varOmega |}\int _\varOmega f(x)dx$.

Lemma 4.10

(Lemma 4.6 of Espejo and Winkler (2018)) Let $\lambda >0$, $C>0$, and suppose that $y\in C^1([0,\infty ))$ and $h\in C^0([0,\infty ))$ are nonnegative functions satisfying $y'(t)+\lambda y(t)\le h(t)$ for some $\lambda >0$ and all $t>0$. Then if $\int _0^\infty h(s)ds\le C$, we have $y(t)\rightarrow 0$ as $t\rightarrow \infty$.

By means of the testing procedure and the Young inequality, we have

(4.3.23)

$$\begin{aligned} \frac{d}{dt}\int _\varOmega (m-\overline{m})^2&=2\int _\varOmega (m-\overline{m})(\varDelta m-u\cdot \nabla m-\rho m+\overline{\rho m}) \\&=2\int _\varOmega m(\varDelta m-u\cdot \nabla m)-2\int _\varOmega (m-\overline{m})(\rho m-\overline{\rho m})\nonumber \\&\le -2\int _\varOmega |\nabla m|^2-2\int _\varOmega (m-\overline{m})\rho m,\nonumber \end{aligned}$$

(4.3.24)

$$\begin{aligned} \frac{d}{dt}\int _\varOmega (c-\overline{c})^2&=2\int _\varOmega (c-\overline{c})(\varDelta c-u\cdot \nabla c-(c-\overline{c})+(m-\overline{m})) \\&=2\int _\varOmega c(\varDelta c-u\cdot \nabla c)-2\int _\varOmega (c-\overline{c})^2+2\int _\varOmega (c-\overline{c})(m-\overline{m})\nonumber \\&\le -2\int _\varOmega |\nabla c|^2-\int _\varOmega (c-\overline{c})^2+\int _\varOmega (m-\overline{m})^2,\nonumber \end{aligned}$$

(4.3.25)

$$\begin{aligned} \frac{d}{dt}\int _\varOmega |u|^2&=-2\int _\varOmega |\nabla u|^2+2\int _\varOmega (\rho +m)\nabla \phi \cdot u-2\int _\varOmega \nabla P\cdot u \\&=-2\int _\varOmega |\nabla u|^2+2\int _\varOmega (\rho -\overline{\rho }+m-\overline{m})\nabla \phi \cdot u\nonumber \\&\le -2\int _\varOmega |\nabla u|^2+K_2\left( \int _\varOmega |\rho -\overline{\rho }+m-\overline{m}|^2\right) ^{\frac{1}{2}}\left( \int _\varOmega |u|^2\right) ^{\frac{1}{2}}\nonumber \\&\le -\int _\varOmega |\nabla u|^2+K_3\left( \int _\varOmega |\rho -\overline{\rho }|^2+\int _\varOmega |m-\overline{m}|^2\right) \nonumber , \end{aligned}$$

(4.3.26)

where $\nabla \cdot u=0$, $u\mid _{\partial \varOmega }=0$ and the boundedness of $u,\nabla \phi$ and $\mathscr {S}$ are used.

Lemma 4.11

Under the assumptions of Lemma 4.6,

$$\begin{aligned}&\Vert (\rho -\overline{\rho })(\cdot ,t)\Vert _{L^\infty (\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty , \\&\Vert (m-\overline{m})(\cdot ,t)\Vert _{L^\infty (\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty , \\&\Vert (c-\overline{c})(\cdot ,t)\Vert _{L^\infty (\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty , \\&\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty . \end{aligned}$$

Proof

From (4.3.23)–(4.3.26), it follows that

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega (\rho -\overline{\rho })^2\le -\int _\varOmega |\nabla \rho |^2+K_1\int _\varOmega |\nabla c|^2+2\overline{\rho }\int _\varOmega \rho m, \end{aligned}$$

(4.3.27)

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega (m-\overline{m})^2\le -2\int _\varOmega |\nabla m|^2+2\overline{m}\int _\varOmega \rho m, \end{aligned}$$

(4.3.28)

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega (c-\overline{c})^2\le -2\int _\varOmega |\nabla c|^2-\int _\varOmega (c-\overline{c})^2+\int _\varOmega (m-\overline{m})^2, \end{aligned}$$

(4.3.29)

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega |u|^2\le -\int _\varOmega |\nabla u|^2+K_3\left( \int _\varOmega |\rho -\overline{\rho }|^2+\int _\varOmega |m-\overline{m}|^2\right) . \end{aligned}$$

(4.3.30)

Since $\int _\varOmega |m-\overline{m}|^2\le C_p \Vert \nabla m\Vert _{L^2(\varOmega )}^2$ and $\int _0^\infty \int _\varOmega \rho m\le K_4$ by (4.3.3), an application of Lemma 4.10 to (4.3.28) yields

$$\begin{aligned} \Vert m(\cdot ,t)-\overline{m}(t)\Vert _{L^2(\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty . \end{aligned}$$

(4.3.31)

Since

$$\begin{aligned} \int _0^\infty \int _\varOmega |(m-\overline{m})|^2ds\le C_p\int _0^\infty \Vert \nabla m\Vert _{L^2(\varOmega )}^2ds \le K_5, \end{aligned}$$

(4.3.32)

the application of Lemma 4.10 to (4.3.29) also yields

$$\begin{aligned} \Vert c(\cdot ,t)-\overline{c}(t)\Vert _{L^2(\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty \end{aligned}$$

(4.3.33)

and

$$\begin{aligned} \int _0^\infty \Vert \nabla c\Vert _{L^2(\varOmega )}^2\le \int _0^\infty \int _\varOmega |m-\overline{m}|^2+\int _\varOmega |c_0-\overline{c_0}|^2 \le K_6. \end{aligned}$$

(4.3.34)

Furthermore, by (4.3.34), $\int _\varOmega |\rho -\overline{\rho }|^2\le C_p \Vert \nabla \rho \Vert _{L^2(\varOmega )}^2$ and $\int _0^\infty \int _\varOmega \rho m\le K_4$, Lemma 4.10 implies that

$$\begin{aligned}&\Vert \rho (\cdot ,t)-\overline{\rho }(t)\Vert _{L^2(\varOmega )}\rightarrow 0 \quad \hbox {as}~t\rightarrow \infty , \end{aligned}$$

(4.3.35)

$$\begin{aligned}&\int _0^\infty \Vert \rho -\overline{\rho }\Vert _{L^2(\varOmega )}^2\le C_p\int _0^\infty \Vert \nabla \rho \Vert _{L^2(\varOmega )}^2\le K_7. \end{aligned}$$

(4.3.36)

Hence, from (4.3.32), (4.3.36), $\int _\varOmega |u|^2\le C_p \Vert \nabla u\Vert _{L^2(\varOmega )}^2$ and Lemma 4.10, it follows that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^2(\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty \end{aligned}$$

(4.3.37)

as well as $\int _0^\infty \Vert \nabla u\Vert _{L^2(\varOmega )}^2\le K_8.$

Now we turn the above convergence in $L^2(\varOmega )$ into $L^\infty (\varOmega )$ with the help of the higher regularity of the solutions. Indeed, similar to the proof of $\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le K$ in Theorem 4.1 in the case $\mathscr {S}=0$ on $\partial \varOmega$, $\Vert m(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le K_{10}$ can be proved since $\Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_9$ for all $t>0$ in (4.3.21). Hence, from (4.3.21), there exists a constant $K_{11}>0$, such that $\Vert m(\cdot ,t)-\overline{m}(t)\Vert _{W^{1,\infty }(\varOmega )}\le K_{11},~\Vert c(\cdot ,t)-\overline{c}(t)\Vert _{W^{1,\infty }(\varOmega )}\le K_{11},~\Vert u(\cdot ,t)\Vert _{W^{1,5}(\varOmega )}\le K_{11}$ for all $t>1$. Therefore, by (4.3.31), (4.3.33) and (4.3.37), the application of the interpolation inequality yields as $t\rightarrow \infty$,

$$\begin{aligned}&\Vert m-\overline{m}\Vert _{L^\infty (\varOmega )}\le C\left( \Vert m-\overline{m}\Vert _{W^{1,\infty }(\varOmega )}^{\frac{3}{5}} \Vert m-\overline{m}\Vert _{L^2(\varOmega )}^{\frac{2}{5}}+\Vert m-\overline{m}\Vert _{L^2(\varOmega )}\right) \rightarrow 0,\\&\Vert c(\cdot ,t)-\overline{c}(t)\Vert _{L^\infty (\varOmega )}\rightarrow 0,\quad \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\rightarrow 0. \end{aligned}$$

In addition, similar to Lemma 4.4 in Espejo and Winkler (2018) or Lemma 5.2 in Cao and Lankeit (2016), there exist $\vartheta \in (0,1)$ and constant $K_{12}>0$ such that $\Vert \rho \Vert _{C^{\vartheta ,\frac{\vartheta }{2}}(\overline{\varOmega }\times [t,t+1])}\le K_{12}$ for all $t>1$, which along with (4.3.35) implies that $\Vert \rho (\cdot ,t)-\overline{\rho }(t)\Vert _{C_{loc}(\overline{\varOmega })} \rightarrow 0 \quad \hbox {as}~t\rightarrow \infty$ and then by the finite covering theorem, $\Vert \rho (\cdot ,t)-\overline{\rho }(t)\Vert _{L^\infty ({\varOmega })} \rightarrow 0 \quad \hbox {as}~t\rightarrow \infty .$

By a very similar argument as in Lemma 4.2 of Espejo and Winkler (2018), we have

Lemma 4.12

Under the assumptions of Lemma 4.6,

$$\begin{aligned} \overline{\rho }(t)\rightarrow \rho _\infty , \quad \overline{m}(t)\rightarrow m_\infty ,\quad \overline{c}(t)\rightarrow m_\infty \quad \hbox {as}~t\rightarrow \infty \end{aligned}$$

with $\rho _\infty =\{\overline{\rho _0}-\overline{m_0}\}_+$ and $m_\infty =\{\overline{m_0}-\overline{\rho _0}\}_+$.

Proof

From (4.3.3) and (4.3.5), we have

$$\begin{aligned}&\int _{t-1}^t\Vert \rho m\Vert _{L^1(\varOmega )}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty , \end{aligned}$$

(4.3.38)

$$\begin{aligned}&\int _{t-1}^t\Vert \nabla m\Vert _{L^2(\varOmega )}^2\rightarrow 0\quad \hbox {as}~t\rightarrow \infty . \end{aligned}$$

(4.3.39)

On the other hand,

$$\begin{aligned} \int _{t-1}^t\Vert \rho m\Vert _{L^1(\varOmega )}&=\int _{t-1}^t\int _\varOmega \rho (m-\overline{m}) +\int _{t-1}^t\int _\varOmega \rho \overline{m} \\&\ge -\int _{t-1}^t\Vert \rho (\cdot ,s)\Vert _{L^2(\varOmega )}\Vert m-\overline{m}\Vert _{L^2(\varOmega )}+|\varOmega |\int _{t-1}^t\overline{\rho }\cdot \overline{m} \\&\ge -K\int _{t-1}^t\Vert \nabla m\Vert _{L^2(\varOmega )}+|\varOmega |\int _{t-1}^t\overline{\rho }\cdot \overline{m} \\&\ge -K\left( \int _{t-1}^t\Vert \nabla m\Vert _{L^2(\varOmega )}^2\right) ^{\frac{1}{2}}+|\varOmega |\int _{t-1}^t\overline{\rho }\cdot \overline{m}. \end{aligned}$$

Inserting (4.3.38) and (4.3.39) into the above inequality, we obtain

$$\begin{aligned} \int _{t-1}^t\overline{\rho }\cdot \overline{m}\rightarrow 0\quad \hbox {as}~t\rightarrow \infty . \end{aligned}$$

(4.3.40)

Now if $\overline{\rho _0}-\overline{m_0}\ge 0$, (4.3.4) warrants that $\overline{\rho }-\overline{m}\ge 0$, which along with (4.3.40) implies that

$$\begin{aligned} \int _{t-1}^t\overline{m}^2(s)ds\rightarrow 0\quad \hbox {as}~t\rightarrow \infty . \end{aligned}$$

(4.3.41)

Noticing that $\overline{m}(s)\ge \overline{m}(t)~\hbox {for all}~t\ge s$, we have $0\le \overline{m}(t)^2\le \int _{t-1}^t \overline{m}^2(s) ds \rightarrow 0\quad \hbox {as}~t\rightarrow \infty ,$ and thus $\overline{\rho }\rightarrow \rho _\infty ~\hbox {as}~t\rightarrow \infty$ due to (4.3.4). By very similar argument, one can see that $\overline{\rho }\rightarrow 0 ~\hbox {as}~ t\rightarrow \infty$ and $\overline{m}\rightarrow m_\infty ~\hbox {as}~t\rightarrow \infty$ in the case of $\overline{\rho _0}-\overline{m_0}<0$. Finally, it is observed that $c(\cdot ,t)\rightarrow m_\infty ~\hbox {in}~ L^2(\varOmega )\,\,\hbox {as}~ t\rightarrow \infty$ is also valid (see Lemma 4.7 of Espejo and Winkler (2018) for example) and thus $\overline{c}(t)\rightarrow m_\infty ~\hbox {as}~t\rightarrow \infty$ by the Hölder inequality.

Combining Lemma 4.11 with Lemma 4.12, we have

Lemma 4.13

Under the assumptions of Lemma 4.6, we have

$$\rho (\cdot ,t)\rightarrow \rho _\infty ,\;m(\cdot ,t)\rightarrow m_\infty ,\; c(\cdot ,t)\rightarrow m_\infty ,\; u(\cdot ,t)\rightarrow 0~~ \hbox {in}~ L^\infty (\varOmega ) \,\, \hbox {as}\,\,t\rightarrow \infty . $$

Now we proceed to estimate the decay rate of $\Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}$, $\Vert m(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}$, $\Vert c(\cdot ,t)-c_\infty \Vert _{L^\infty (\varOmega )}$, and $\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ when $\int _\varOmega \rho _0\ne \int _\varOmega m_0$. To this end, we first consider its decay rate in $L^2(\varOmega )$ based on a differential inequality.

Lemma 4.14

Under the assumptions of Lemma 4.6 and $\int _\varOmega \rho _0\ne \int _\varOmega m_0$, for any $\varepsilon >0$, there exist constants $K(\varepsilon )>0$ and $t_\varepsilon >0$ such that for $t>t_\varepsilon$,

$$\begin{aligned}&|\overline{\rho }(t)-\rho _\infty |+|\overline{m}(t)-m_\infty |\le&K(\varepsilon ) e^{-(\rho _\infty +m_\infty -\varepsilon )t}, \end{aligned}$$

(4.3.42)

$$\begin{aligned}&|\overline{c}(t)-m_\infty |\le&K(\varepsilon ) e^{-\min \{1,(\rho _\infty +m_\infty -\varepsilon )\}t}. \end{aligned}$$

(4.3.43)

Proof

For the case $\int _\varOmega \rho _0>\int _\varOmega m_0$, we have $\rho _\infty >0$ and $m_\infty =0$. By Lemma 4.13, there exists $t_\varepsilon >0$ such that $\rho (x,t)\ge \rho _\infty -\varepsilon$ for $t>t_\varepsilon$ and $x\in \varOmega$, and thereby $\frac{d}{dt}\int _\varOmega m=-\int _\varOmega \rho m\le -(\rho _\infty -\varepsilon )\int _\varOmega m$ for $t>t_\varepsilon$, which implies that $\overline{m}(t)\le \overline{m_0} e^{-(\rho _\infty -\varepsilon ) (t-t_\varepsilon )}$ for $t>t_\varepsilon$. Moreover, due to $\overline{\rho }=\overline{m}+\rho _\infty$ by (4.3.4), we have $|\overline{\rho }(t)-\rho _\infty |=\overline{m}(t)\le \overline{m_0}e^{-(\rho _\infty -\varepsilon ) (t-t_\varepsilon )}\quad \hbox {for}~t>t_\varepsilon .$ As for the case $\int _\varOmega \rho _0<\int _\varOmega m_0$, similarly we can prove that $|\overline{m}(t)-m_\infty |= \overline{\rho }\le \overline{\rho _0}e^{-(m_\infty -\varepsilon ) (t-t_\varepsilon )}.$ for $t>t_\varepsilon$. Furthermore, by the third equation of (4.1.1), we have $\frac{d}{dt}\int _\varOmega (c-m_\infty )=\int _\varOmega (m-m_\infty )-\int _\varOmega (c-m_\infty ),$ and thereby $|\overline{c}(t)-m_\infty |\le K(\varepsilon )e^{-\min \{1,\rho _\infty +m_\infty -\varepsilon \}t}.$

Proof of Theorem 4.2 in the case $\mathscr {S}=0$ on $\partial \varOmega$. By Lemmas 4.11 and 4.13, as $t\rightarrow \infty$, we have

$$\begin{aligned} \rho (\cdot ,t)-\overline{\rho }(t)\rightarrow 0, ~ m(\cdot ,t)-\overline{m}(t)\rightarrow 0, ~ \rho (\cdot ,t)\rightarrow \rho _\infty , ~ m(\cdot ,t)\rightarrow m_\infty \quad \hbox {in}~L^\infty (\varOmega ), \end{aligned}$$

which implies that for any $\varepsilon \in (0,\frac{\rho _\infty + m_\infty }{2} )$, there exists $t_\varepsilon >0$ such that $|\rho (\cdot ,t)-\overline{\rho }(t)|<\varepsilon$, $|m(\cdot ,t)-\overline{m}(t)|<\varepsilon$, $\rho (\cdot ,t)+m(\cdot ,t)\ge \rho _\infty + m_\infty -\varepsilon$ for all $t>t_\varepsilon$ and $x\in \varOmega$. Hence, from (4.3.23)–(4.3.26), we have

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega (\rho -\overline{\rho })^2+\int _\varOmega |\nabla \rho |^2\le K_1\int _\varOmega |\nabla c|^2+2\varepsilon \int _\varOmega \rho m, \end{aligned}$$

(4.3.44)

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega (m-\overline{m})^2+2\int _\varOmega |\nabla m|^2\le 2\varepsilon \int _\varOmega \rho m, \end{aligned}$$

(4.3.45)

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega (c-\overline{c})^2+2\int _\varOmega |\nabla c|^2+\int _\varOmega (c-\overline{c})^2\le \int _\varOmega (m-\overline{m})^2, \end{aligned}$$

(4.3.46)

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega |u|^2+\int _\varOmega |\nabla u|^2\le K_3\left( \int _\varOmega (\rho -\overline{\rho })^2+\int _\varOmega (m-\overline{m})^2\right) \end{aligned}$$

(4.3.47)

for $t>t_\varepsilon$, as well as

$$\begin{aligned}&\frac{d}{dt}\int _\varOmega \rho m \\ =&\int _\varOmega \left[ \rho (\varDelta m-u\cdot \nabla m-\rho m)+m(\varDelta \rho -\nabla (\rho S(x,\rho ,c)\nabla c)-u\cdot \nabla \rho -\rho m)\right] \nonumber \\ =&-2\int _\varOmega \nabla \rho \nabla m-\int _\varOmega (\rho u\cdot \nabla m+m u\cdot \nabla \rho )+\int _\varOmega \rho S(x,\rho ,c)\nabla c \cdot \nabla m-\int _\varOmega \rho m^2\nonumber \\&-\int _\varOmega \rho ^2\,m\nonumber \\ \le&\int _\varOmega |\nabla \rho |^2+2\int _\varOmega |\nabla m|^2-\int _\varOmega u\cdot \nabla (\rho m)+K_3\int _\varOmega |\nabla c|^2-\int _\varOmega \rho m(\rho +m)\nonumber \\ \le&\int _\varOmega |\nabla \rho |^2+2\int _\varOmega |\nabla m|^2+K_3\int _\varOmega |\nabla c|^2-\frac{1}{2}(\rho _\infty +m_\infty )\int _\varOmega \rho m,\nonumber \end{aligned}$$

(4.3.48)

where $\nabla \cdot u=0$, $u\mid _{\partial \varOmega }=0$ and the boundedness of $\rho$ are used.

On the other hand, by Poincare’s inequality, there exists $C_P>0$, such that

$$\begin{aligned}&\int _\varOmega |\nabla \rho |^2\ge C_{P} \int _\varOmega (\rho -\overline{\rho })^2,\quad \int _\varOmega |\nabla m|^2\ge C_{P}\int _\varOmega (m-\overline{m})^2, \\&\int _\varOmega |\nabla c|^2\ge C_{P}\int _\varOmega (c-\overline{c})^2,\quad \int _\varOmega |\nabla u|^2\ge C_{P}\int _\varOmega (u-\overline{u})^2. \end{aligned}$$

Therefore, combining the above inequalities, and taking $\varepsilon <\frac{ a(\rho _\infty +m_\infty )C_P}{8(K_1+C_P)}$ with $a=\min \{\frac{1}{2},\frac{K_1}{4C_P},\frac{K_1}{K_3}\}$, the functional $G(t):=\int _\varOmega (\rho -\overline{\rho })^2+\frac{K_1}{C_P}\int _\varOmega (m-\overline{m})^2+K_1\int _\varOmega (c-\overline{c})^2 +a\int _\varOmega \rho m$ satisfies the ordinary differential inequality $\frac{d}{dt} G(t)+ \delta _1 G(t)\le 0$ with $\delta _1=\min \{\frac{C_P}{2}, 1,\frac{\rho _\infty +m_\infty }{4}\}$, which implies that

$$\begin{aligned} \Vert \rho (\cdot ,t)-\overline{\rho }\Vert _{L^2(\varOmega )}+ \Vert m(\cdot ,t)-\overline{m}\Vert _{L^2(\varOmega )}+ \Vert c(\cdot ,t)-\overline{c}\Vert _{L^2(\varOmega )}\le Ce^{-\frac{\delta _1}{2}t}. \end{aligned}$$

(4.3.49)

Moreover, by (4.3.49) and (4.3.47), $\Vert u(\cdot ,t)\Vert _{L^2(\varOmega )}\le Ce^{-\delta _2 t}$ for some $\delta _2>0$. At this position, combining (4.3.49) with Lemma 4.14, we can find $\delta _3>0$ such that

$$\begin{aligned} \Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^2(\varOmega )}+ \Vert m(\cdot ,t)-m_\infty \Vert _{L^2(\varOmega )}+ \Vert c(\cdot ,t)-m_\infty \Vert _{L^2(\varOmega )}\le Ce^{-\delta _3 t}. \end{aligned}$$

(4.3.50)

Hence, as in the proof of Lemma 4.11, we can obtain the decay estimates (4.1.9)–(4.1.12) by an application of the interpolation inequality, and thus the proof is complete.

3. Exponential decay under smallness condition

In this subsection, we give the proof of Theorem 4.3 under the assumption that $\mathscr {S}=0$ on $\partial \varOmega$. The proof thereof is divided into two cases (Propositions 4.1 and 4.2).

(1) The case $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$

In this subsection, we consider the case $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$, i.e., $\rho _\infty >0$, $m_\infty =0$.

Proposition 4.1

Suppose that (4.1.4) hold with $\alpha =0$ and $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$. Let $N=3$, $p_0\in (\frac{N}{2}, N)$, $q_0\in (N,\frac{Np_0}{N-p_0})$. There exists $\varepsilon >0$ such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.7) as well as

$$\begin{aligned} \Vert \rho _0-\rho _\infty \Vert _{L^{p_0}(\varOmega )}\le \varepsilon ,\quad \Vert m_0\Vert _{L^{q_0}(\varOmega )}\le \varepsilon , \quad \Vert \nabla c_0\Vert _{L^{N}(\varOmega )}\le \varepsilon , \quad \Vert u_0\Vert _{L^{N}(\varOmega )}\le \varepsilon , \end{aligned}$$

(4.1.1) admits a global classical solution $(\rho ,m,c,u,P)$. In particular, for any $\alpha _1\in (0,\min \{\lambda _1,\rho _\infty \})$, $\alpha _2\in (0,\min \{\alpha _1,\lambda _1',1\})$, there exist constants $K_i$, $i=1,2,3,4$, such that for all $t\ge 1$

$$\begin{aligned}&\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t}, \end{aligned}$$

(4.3.51)

$$\begin{aligned}&\Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \end{aligned}$$

(4.3.52)

$$\begin{aligned}&\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \end{aligned}$$

(4.3.53)

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

(4.3.54)

Proposition 4.1 is the consequence of the following lemmas. In the proof of these lemmas, the constants $c_i>0$, $i=1,\ldots ,10$, refer to those in Lemmas 1.1, 4.1–4.3, respectively. We first collect some easily verifiable observations in the following lemma:

Lemma 4.15

Under the assumptions of Proposition 4.1 and

$$\sigma =\int _0^\infty \left( 1+s^{-\frac{N}{2p_0}}\right) e^{-\alpha _1s}ds,$$

there exist $M_1>0,M_2>0$ and $\varepsilon >0$, such that

$$\begin{aligned}&c_3+2 c_2c_{10} e^{(1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma } \le \displaystyle \frac{M_2}{4}, \quad M_1 \varepsilon <1, \end{aligned}$$

(4.3.55)

$$\begin{aligned}&12 c_2 c_{10} (c_6+ 4c_6c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}(M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}\nonumber \\&\quad +4e^{(1+c_1+ c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma })) \varepsilon <1, \end{aligned}$$

(4.3.56)

$$\begin{aligned}&c_4 c_{10} C_SM_2 (e^{(1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma }+\rho _\infty |\varOmega |^{\frac{1}{q_0}})\le \displaystyle \frac{M_1}{8}, \end{aligned}$$

(4.3.57)

$$\begin{aligned}&3 c_{10} c_4C_S(M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})M_2\varepsilon \le \displaystyle \frac{M_1}{8}, \end{aligned}$$

(4.3.58)

$$\begin{aligned}&3 c_{10}c_7 c_6(M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}) (1+2c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\nonumber \\&\quad \cdot (M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+4 e^{(1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma }))\varepsilon \le \displaystyle \frac{M_1}{4}. \end{aligned}$$

(4.3.59)

Let

$$\begin{aligned} \!\!T\!\triangleq \!\sup \!\left\{ \!\widetilde{T}\!\in \!(0,T_{max})\!\left| \begin{aligned}&\!\Vert (\rho \!-\!m)(\cdot ,t)\!-\!e^{t\varDelta }(\rho _0\!-\!m_0)\Vert _{L^{\theta }(\varOmega )}\!\\ \le&\! M_1\varepsilon (1\!+\!t^{-\frac{N}{2} (\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}\quad \hbox {for all} \,\,\theta \in [q_0,\infty ], ~t\in [0,\widetilde{T}); \\&\!\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le M_2\varepsilon (1+t^{-\frac{1}{2}})e^{-\alpha _1t}~\hbox {for all}~t\in [0,\widetilde{T}). \end{aligned} \!\!\right. \right\} \end{aligned}$$

(4.3.60)

By (4.1.7) and Lemma 4.4, $T>0$ is well-defined. We first show $T=T_{max}$. To this end, we will show that all of the estimates mentioned in (4.3.60) is valid with even smaller coefficients on the right-hand side. The derivation of these estimates will mainly rely on $L^p-L^q$ estimates for the Neumann heat semigroup and the fact that the classical solutions on $(0,T_{max})$ can be represented as

$$\begin{aligned} (\rho -m)(\cdot ,t) =&e^{t\varDelta }(\rho _0-m_0)\nonumber \\&- \int _{0}^{t}e^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)+ u\cdot \nabla (\rho -m))(\cdot ,s)ds, \end{aligned}$$

(4.3.61)

$$\begin{aligned} m(\cdot ,t)=&e^{t\varDelta }m_0-\int _0^t e^{(t-s)\varDelta }(\rho m+u\cdot \nabla m)(\cdot ,s)ds, \end{aligned}$$

(4.3.62)

$$\begin{aligned} c(\cdot ,t)=&e^{t(\varDelta -1)}c_0+\int _0^t e^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)ds, \end{aligned}$$

(4.3.63)

$$\begin{aligned} u(\cdot ,t)=&e^{-tA}u_0+\int _0^t e^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi )(\cdot ,s)ds \end{aligned}$$

(4.3.64)

for all $t\in (0,T_{max})$ as per the variation-of-constants formula.

Lemma 4.16

Under the assumptions of Proposition 4.1, for all $t\in (0,T)$ and $\theta \in [q_0,\infty ]$,

$$\begin{aligned} \Vert (\rho -m)(\cdot ,t)-\rho _\infty \Vert _{L^\theta (\varOmega )}\le M_3\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}. \end{aligned}$$

Proof

Since $e^{t\varDelta }\rho _\infty =\rho _\infty$ and $\int _\varOmega (\rho _0-m_0-\rho _\infty )=0$, the definition of T and Lemma 1.1(i) show that

$$\begin{aligned}&\Vert (\rho -m)(\cdot ,t)-\rho _\infty \Vert _{L^\theta (\varOmega )} \\ \le&\Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )} +\Vert e^{t\varDelta }(\rho _0-m_0-\rho _\infty )\Vert _{L^\theta (\varOmega )} \\ \le&M_1\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} +c_1(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})(\Vert \rho _0-\rho _\infty \Vert _{L^{p_0}(\varOmega )} \\&+\Vert m_0\Vert _{L^{p_0}(\varOmega )})e^{-\lambda _1t} \\ \le&M_3\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} \end{aligned}$$

for all $t\in (0,T)$ and $\theta \in [q_0,\infty ]$, where $M_3=M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}$.

Lemma 4.17

Under the assumptions of Proposition 4.1, for any $k>1$,

$$\begin{aligned} \Vert m(\cdot ,t)\Vert _{L^k(\varOmega )}\le M_4 \Vert m_0\Vert _{L^k(\varOmega )} e^{-\rho _\infty t}\quad \hbox {for all}~ t\in (0,T) \end{aligned}$$

(4.3.65)

with $\sigma =\int _0^\infty (1+s^{-\frac{N}{2p_0}})e^{-\alpha _1s}ds$ and $M_4=e^{M_3\sigma \varepsilon }$.

Proof

Multiplying the m-equation in (4.1.1) by $km^{k-1}$ and integrating the result over $\varOmega$, we get $\frac{d}{dt}\int _\varOmega m^k\le -k\int _\varOmega \rho m^k$ on (0, T). Since

$$-\rho \le |\rho -m-\rho _\infty |-m-\rho _\infty \le -\rho _\infty +|\rho -m-\rho _\infty |,$$

Lemma 4.16 yields

$$\begin{aligned} \frac{d}{dt}\int _\varOmega m^k&\le -k\rho _\infty \int _\varOmega m^k+k\int _\varOmega m^k|\rho -m-\rho _\infty | \\&\le -k\rho _\infty \int _\varOmega m^k+k\Vert \rho -m-\rho _\infty \Vert _{L^\infty (\varOmega )}\int _\varOmega m^k \\&\le -k\rho _\infty \int _\varOmega m^k+kM_3\varepsilon \left( 1+t^{-\frac{N}{2p_0}}\right) e^{-\alpha _1t}\int _\varOmega m^k \end{aligned}$$

and thus

$$\begin{aligned} \int _\varOmega m^k \le&\int _\varOmega m_0^k \exp \{-k\rho _\infty t+kM_3\varepsilon \int _0^t(1+s^{-\frac{N}{2p_0}})e^{-\alpha _1s}ds\} \\ \le&\Vert m_0\Vert _{L^k(\varOmega )} ^ke^{k(M_3\sigma \varepsilon -\rho _\infty t)}. \end{aligned}$$

The assertion (4.3.65) follows immediately.

Lemma 4.18

Under the assumptions of Proposition 4.1, there exists $M_3>0$, such that $\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le M_5\varepsilon \left( 1+t^{-\frac{1}{2}+\frac{N}{2q_0}}\right) e^{-\alpha _2 t}$ for all $t\in (0,T)$.

Proof

For any given $\alpha _2<\lambda _1'$, we fix $\mu \in (\alpha _2, \lambda _1')$. By (4.3.64), Lemmas 4.1 and 4.2, we obtain

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\nonumber \\ \le&c_6t^{-\frac{N}{2}\left( \frac{1}{N}-\frac{1}{q_0}\right) }e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} +\int _0^t\Vert e^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&c_6t^{-\frac{N}{2}\left( \frac{1}{N}-\frac{1}{q_0}\right) }e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} \\&+c_6\int _0^te^{-\mu (t-s)}\Vert \mathscr {P}((\rho +m-\overline{\rho +m})\nabla \phi )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&c_6 t^{-\frac{1}{2}+\frac{N}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )}\nonumber \\&+c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\int _0^te^{-\mu (t-s)}\Vert (\rho +m-\overline{\rho +m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds,\nonumber \end{aligned}$$

(4.3.66)

where $\mathscr {P}(\overline{\rho +m}\nabla \phi )=\overline{\rho +m} \mathscr {P}(\nabla \phi )=0$ is used. On the other hand, due to $\alpha _1<\rho _\infty$, Lemmas 4.16 and 4.17 show that

$$\begin{aligned}&\Vert (\rho +m-\overline{\rho +m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\nonumber \\ =&\Vert (\rho -m-\overline{\rho -m})(\cdot ,s)+2(m-\overline{m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )} \\ \le&\Vert (\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+2\Vert (m-\overline{m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\nonumber \\ \le&M_5' \varepsilon (1+s^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s} \nonumber \end{aligned}$$

(4.3.67)

with $M_5'=M_3+4e^{M_3 \sigma \varepsilon }$. Combining (4.3.66) with (4.3.67) and applying Lemma 4.2, we have

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )} \\ \le&c_6t^{-\frac{1}{2}+\frac{N}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} \\&+c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )} M_5'\varepsilon \int _0^t(1+s^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s}e^{-\mu (t-s)}ds \\ \le&c_6 t^{-\frac{1}{2}+\frac{N}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} +c_6c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}M_5'\varepsilon (1+t^{\min \{0,1-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}})e^{-\alpha _2t} \\ \le&c_6 t^{-\frac{1}{2}+\frac{N}{2q_0}}e^{-\mu t}\varepsilon +2 c_6c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}M_5'\varepsilon e^{-\alpha _2t} \\ \le&M_5\varepsilon (1+t^{-\frac{1}{2}+\frac{N}{2q_0}})e^{-\alpha _2t}, \end{aligned}$$

where $M_5=c_6+ 2 c_6c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}M_5'$ and $\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})<1$ is used.

Lemma 4.19

Under the assumptions of Proposition 4.1, for all $t\in (0,T)$,

$$\begin{aligned} \Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le \frac{M_2}{2}\varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _1 t}. \end{aligned}$$

Proof

By (4.3.63) and Lemma 1.1(iii), we have

$$\begin{aligned}&\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\nonumber \\ \le&\Vert e^{t(\varDelta -1)}\nabla c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\ \le&c_3(1+t^{-\frac{1}{2}})e^{-(\lambda _1+1)t}\Vert \nabla c_0\Vert _{L^N(\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}m(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds.\nonumber \end{aligned}$$

(4.3.68)

Now we estimate the last two integrals on the right-hand side of the above inequality. From Lemmas 1.1(ii), 4.3, 4.17 with $k=q_0$ and the fact that $q_0>N$, it follows that

$$\begin{aligned}&\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}m\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert m\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_2 M_4\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)} e^{-\rho _\infty s}ds \nonumber \\ \le&c_2c_{10} M_4 (1+t^{\min \{0,\frac{1}{2}-\frac{N}{2q_0}\}})\varepsilon e^{-\alpha _1t}\nonumber \\ \le&2 c_2c_{10} M_4(1+t^{-\frac{1}{2}})\varepsilon e^{-\alpha _1 t}.\nonumber \end{aligned}$$

(4.3.69)

On the other hand, by Lemmas 4.3, 4.18 and the definition of T, we obtain

$$\begin{aligned}&\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u\cdot \nabla c\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2 M_5M_2 \varepsilon ^2\!\!\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)}(1+s^{-\frac{1}{2}+\frac{N}{2q_0}}) (1+s^{-\frac{1}{2}}) e^{-(\alpha _1+\alpha _2) s}ds \nonumber \\ \le&3c_2M_5M_2\varepsilon ^{2}\int _0^te^{-(\lambda _1+1)(t-s)} e^{-(\alpha _1+\alpha _2) s}(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})(1+s^{-1+\frac{N}{2q_0}})ds\nonumber \\ \le&3c_2 c_{10}M_2 M_5\varepsilon ^2(1+t^{-\frac{1}{2}})e^{-\alpha _1 t}.\nonumber \end{aligned}$$

(4.3.70)

From (4.3.68)–(4.3.70), it follows that

$$\begin{aligned} \Vert \nabla c\Vert _{L^\infty (\varOmega )}&\le (c_3+2 c_2c_{10} M_4 +3c_2 c_{10}M_2 M_5\varepsilon ) (1+t^{-\frac{1}{2}})\varepsilon e^{-\alpha _1t} \\&\le \frac{M_2}{2}(1+t^{-\frac{1}{2}})\varepsilon e^{-\alpha _1 t}, \end{aligned}$$

due to the choice of $M_1,M_2$ and $\varepsilon$ satisfying (4.3.55), (4.3.56), and thereby completes the proof.

Lemma 4.20

Under the assumptions of Proposition 4.1, for all $\theta \in [q_0,\infty ]$ and $t\in (0,T)$,

$$\begin{aligned} \Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) e^{-\alpha _1 t}. \end{aligned}$$

Proof

According to (4.3.61), Lemmas 1.1(iv) and 4.1, we have

$$\begin{aligned}&\Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )} \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)+u\cdot \nabla (\rho -m))(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot ((\rho -m-\rho _\infty )u)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&c_4C_S\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\&+c_7\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\mu (t-s)}\Vert u(\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ =&I_1+I_2. \end{aligned}$$

Now we need to estimate $I_1$ and $I_2$. Firstly, from Lemmas 4.16 and 4.17, we obtain

$$\begin{aligned} \Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}&\le \Vert (\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+\Vert m(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+\Vert \rho _\infty \Vert _{L^{q_0}(\varOmega )}\nonumber \\&\le M_3\varepsilon (1+s^{-\frac{N}{2}\left( \frac{1}{p_0}-\frac{1}{q_0}\right) })e^{-\alpha _1s}+M_6 \end{aligned}$$

(4.3.71)

with $M_6=e^{(1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma }+\rho _\infty |\varOmega |^{\frac{1}{q_0}}$, which together with Lemmas 4.19 and 1.1 implies that

$$\begin{aligned} I_1\le&c_4C_SM_6\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \\&+ M_7\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })})(1+s^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-\alpha _1s} e^{-\lambda _1(t-s)}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \nonumber \\ \le&c_4C_SM_6M_2\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}(1+s^{-\frac{1}{2}}) e^{-\alpha _1 s}ds \nonumber \\&+3 M_7M_2\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })})(1+s^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-2\alpha _1s} e^{-\lambda _1(t-s)}ds\nonumber \\ \le&c_{10} (c_4C_SM_6M_2+3 M_7M_2\varepsilon )(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})\varepsilon e^{-\alpha _1 t}\nonumber \\ \le&\frac{M_1}{4}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1 t}\nonumber \end{aligned}$$

(4.3.72)

with $M_7:=c_4C_SM_3$, where we have used (4.3.57) and (4.3.58) and $\frac{1}{p_0}-\frac{1}{q_0}<\frac{1}{N}$. On the other hand, from Lemmas 4.16 and 4.18, it follows that

$$\begin{aligned} I_2&=c_7\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\mu (t-s)}\Vert \rho -m-\rho _\infty \Vert _{L^\infty (\varOmega )}\Vert u\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&\le 3c_7M_3M_5\varepsilon ^{2}\!\!\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\mu (t-s)}(1+s^{-\frac{1}{2}+\frac{N }{ 2q_0}-\frac{N}{2p_0}})e^{-(\alpha _1+\alpha _2)s} ds\nonumber \\&\le 3c_7M_3M_5c_{10} \varepsilon ^{2}(1+t^{\min \{0,\frac{N}{2} (\frac{1}{\theta }-\frac{1}{p_0})\}})e^{-\min \{\mu ,\alpha _1+\alpha _2\}t }\nonumber \\&\le \frac{M_1}{4}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1 t}, \end{aligned}$$

(4.3.73)

where we have used (4.3.59) and $\frac{1}{p_0}-\frac{1}{q_0}<\frac{1}{N}$. Hence, combining the above inequalities leads to our conclusion immediately.

Proof of Theorem 4.3 in the case $\mathscr {S}=0$ on $\partial \varOmega$, part 1 (Proposition 4.1). First we claim that $T=T_{max}$. In fact, if $T<T_{max}$, then by Lemmas 4.19 and 4.20, we have $\Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le \frac{M_2}{2}\varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _1 t}$ and

$$\begin{aligned} \Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) e^{-\alpha _1 t} \end{aligned}$$

for all $\theta \in [q_0,\infty ]$ and $t\in (0,T)$, which contradicts the definition of T in (4.3.60). Next, we show that $T_{max}=\infty$. In fact, if $T_{max}<\infty$, we only need to show that as $t \rightarrow T_{max}$,

$$ \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}\rightarrow \infty $$

according to the extensibility criterion in Lemma 4.4.

Let $t_0:=\min \{1,\frac{T_{max}}{3}\}$. Then from Lemma 4.17, there exists $K_1>0$ such that for $t\in (t_0,T_{max})$,

$$\begin{aligned} \Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_1e^{-\rho _\infty t}. \end{aligned}$$

(4.3.74)

Moreover, from Lemma 4.16 and the fact that

$$\Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le \Vert (\rho -m)(\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}, $$

it follows that for all $t\in (t_0,T_{max})$ and some constant $K_2>0$,

$$\begin{aligned} \Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}. \end{aligned}$$

(4.3.75)

Furthermore, Lemma 4.19 implies that there exists $K_3'>0$, such that

$$\begin{aligned} \Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le K_3'e^{-\alpha _2t}\quad \hbox {for all}\,\,t\in (t_0,T_{max}). \end{aligned}$$

(4.3.76)

On the other hand, we can conclude that $\Vert c(\cdot ,t)\Vert _{L^\infty (\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}\le C~\hbox {for }~t\in (t_0,T_{max})$. In fact, we first show that there exists a constant $M_9>0$, such that

$$\begin{aligned} \Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}\le M_9 e^{-\alpha _2 t} \end{aligned}$$

(4.3.77)

for $t_0<t<T_{max}$. By (4.3.64), we have

$$\begin{aligned}&\Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )} \\ \le&\Vert A^\beta e^{-tA} u_0\Vert _{L^2(\varOmega )} +\int _{0}^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((\rho +m-\rho _\infty )\nabla \phi )(\cdot ,s)\Vert _{L^2(\varOmega )}ds. \end{aligned}$$

According to Lemma 4.1, $\Vert A^\beta e^{-tA} u_0\Vert _{L^2(\varOmega )}\le c_5 e^{-\mu t}\Vert A^\beta u_0\Vert _{L^2(\varOmega )}$ for all $t\in (0,T_{max})$. On the other hand, from Lemmas 4.1, 4.2, and 4.16, it follows that there exists $\hat{M}>1$, such that

$$\begin{aligned}&\int _0^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((\rho +m-\rho _\infty )\nabla \phi )(\cdot ,s)\Vert _{L^2(\varOmega )}ds \\ \le&c_9c_5\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} \int _0^t e^{-\mu (t-s)}(t-s)^{-\beta } (\Vert (\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )} \\&+2\Vert m(\cdot ,s)\Vert _{L^{q_0}(\varOmega )})ds \\ \le&c_9c_5\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} \hat{M} \int _0^t e^{-\mu (t-s)}(t-s)^{-\beta } (1+s^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s} ds\\ \le&c_5 c_9 c_{10}\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} \hat{M} e^{-\alpha _2 t}(1+t^{\min \{0,1-\beta -\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}})\\ \le&c_5 c_9 c_{10}\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} \hat{M} e^{-\alpha _2 t}(1+t_0^{\min \{0,1-\beta -\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}}) \end{aligned}$$

for $t_0<t<T_{max}$. Hence, combining the above inequalities, we arrive at (4.3.77).

Since $D(A^\beta )\hookrightarrow L^\infty (\varOmega )$ with $\beta \in (\frac{N}{4},1)$, we have

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t} \quad \hbox {for some}\,K_4>0 \,\hbox {and }\, t\in (0,T_{max}). \end{aligned}$$

(4.3.78)

Now we turn to show that there exists $K_3''>0$, such that

$$\begin{aligned} \Vert c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le K_3'' e^{-\alpha _2t}\quad \hbox {for all}\,\, t\in (0,T_{max}). \end{aligned}$$

(4.3.79)

Indeed, from (4.3.63), it follows that

$$\begin{aligned} \Vert c\Vert _{L^\infty (\varOmega )}&\le \Vert e^{t(\varDelta -1)}c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert e^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&\le e^{-t}\Vert c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert e^{(t-s)(\varDelta -1)}m(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\&\quad +\int _0^t\Vert e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds.\nonumber \end{aligned}$$

(4.3.80)

An application of (4.3.65) with $k=\infty$ yields

$$\begin{aligned} \int _0^t\Vert e^{(t-s)(\varDelta -1)}m(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds&\le \int _0^t e^{-(t-s)}\Vert m(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}ds \\&\le \Vert m_0\Vert _{L^\infty (\varOmega )}M_4\int _0^t e^{-(t-s)} e^{-\rho _\infty s} ds\nonumber \\&\le M_4c_{10} e^{-\alpha _2t}.\nonumber \end{aligned}$$

(4.3.81)

On the other hand, from (4.3.78) and (4.3.76), we can see that

$$\begin{aligned} \int _0^t\Vert e^{(t-s)(\varDelta -1)} u\cdot \nabla c\Vert _{L^\infty (\varOmega )}ds&\le \int _0^t e^{-(t-s)}\Vert u\Vert _{L^{\infty }(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \\&\le K_3' K_4 \int _0^t e^{-2\alpha _2 s} e^{-(t-s)}ds \nonumber \\&\le K_3' K_4 c_{10} e^{-\alpha _2 t}. \nonumber \end{aligned}$$

(4.3.82)

Hence, inserting (4.3.81), (4.3.82) into (4.3.80), we arrive at the conclusion (4.3.79). Therefore, we have $T_{max}=\infty$, and the decay estimates in (4.3.51)–(4.3.54) follow from (4.3.74)–(4.3.79), respectively.

(2) The case $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$

In this subsection, we consider the case $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$, i.e., $m_\infty >0$, $\rho _\infty =0$.

Proposition 4.2

Suppose that (4.1.4) hold with $\alpha =0$ and $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$. Let $N=3$, $p_0\in (\frac{2N}{3}, N)$, $q_0\in (N,\frac{Np_0}{2(N-p_0)})$. Then there exists $\varepsilon >0$ such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.7) as well as

$$\Vert \rho _0\Vert _{L^{p_0}(\varOmega )}\le \varepsilon ,\quad \Vert m_0-m_\infty \Vert _{L^{q_0}(\varOmega )}\le \varepsilon , \quad \Vert \nabla c_0\Vert _{L^{N}(\varOmega )}\le \varepsilon , \quad \Vert u_0\Vert _{L^{N}(\varOmega )}\le \varepsilon , $$

(4.1.1) admits a global classical solution $(\rho ,m,c,u,P)$. Furthermore, for any $\alpha _1\!\in \!(0,\min \{\lambda _1,m_\infty \})$, $\alpha _2\!\in \!(0,\min \{\alpha _1,\lambda _1',1\})$, there exist constants $K_i>0$, $i=1,2,3,4$, such that

$$\begin{aligned}&\Vert m(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t}, \end{aligned}$$

(4.3.83)

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \end{aligned}$$

(4.3.84)

$$\begin{aligned}&\Vert c(\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \end{aligned}$$

(4.3.85)

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

(4.3.86)

The proof of Proposition 4.2 proceeds in a parallel fashion to that of Proposition 4.1. However, due to differences in the properties of $\rho$ and m, there are significant differences in the details of their proofs. Thus, for the convenience of the reader, we will give the full proof of Proposition 4.2. The following can be verified easily:

Lemma 4.21

Under the assumptions of Proposition 4.2, it is possible to choose $M_1>0,M_2>0$ and $\varepsilon >0$, such that

$$\begin{aligned}&c_3\le \displaystyle \frac{M_2}{6},\quad c_2 c_{10}(1+ c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+M_1)\le \displaystyle \frac{M_2}{6}, \end{aligned}$$

(4.3.87)

$$\begin{aligned}&18c_2c_6 c_{10} (1+2c_9 c_{10} (1+c_1+ c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+2M_1) \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}) \varepsilon \le 1, \end{aligned}$$

(4.3.88)

$$\begin{aligned}&2c_1+(\min \{1,|\varOmega |\})^{-\frac{1}{p_0}}\le \displaystyle \frac{M_1}{8},\quad 24c_4C_Sc_{10}M_2 \varepsilon <1, \end{aligned}$$

(4.3.89)

$$\begin{aligned}&24c_4c_{10} c_6 ( 1+2c_9 c_{10} (1+c_1+ c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+2M_1) \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}) \varepsilon <1, \end{aligned}$$

(4.3.90)

$$\begin{aligned}&24c_4c_{10}(1+ c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+M_1)\varepsilon <1, \end{aligned}$$

(4.3.91)

$$\begin{aligned}&12c_4C_Sc_{10} M_1M_2 \varepsilon < 1, \end{aligned}$$

(4.3.92)

$$\begin{aligned}&c_{10}c_6 c_4(1+ c_1\!+\!c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}) (1+2c_9 c_{10} (1+c_1+ c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+2M_1) \Vert \nabla \phi \Vert _{L^\infty (\varOmega )})\varepsilon \nonumber \\&<\frac{1}{24}. \end{aligned}$$

(4.3.93)

Similar to the proof of Proposition 4.1, we define

$$\begin{aligned} T\!\triangleq \!\sup \!\left\{ \!\widetilde{T}\!\in \!(0,T_{max})\!\left| \begin{aligned}&\!\Vert (m\!-\!\rho )(\cdot ,t)\!-\!e^{t\varDelta }(m_0\!-\!\rho _0)\Vert _{L^{\theta }(\varOmega )}\!\le \! \varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}, \\&\!\Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )}\le M_1\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t},\,\forall \theta \in [q_0,\infty ], \\&\!\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le M_2\varepsilon (1+t^{-\frac{1}{2}})e^{-\alpha _1 t} ,\,\forall t\in [0,\widetilde{T}). \end{aligned}\right. \!\right\} \end{aligned}$$

(4.3.94)

By Lemma 4.3.7 and (4.1.7), $T>0$ is well-defined. As in the previous subsection, we first show $T=T_{max}$, and then $T_{max}=\infty$. To this end, we will show that all of the estimates mentioned in (4.3.94) are valid with even smaller coefficients on the right-hand side than appearing in (4.3.94). The derivation of these estimates will mainly rely on $L^p-L^q$ estimates for the Neumann heat semigroup and the corresponding semigroup for Stokes operator, and the fact that the classical solutions of (4.1.1) on (0, T) can be represented as

$$\begin{aligned}&(m-\rho )(\cdot ,t)=e^{t\varDelta }(m_0-\rho _0)\nonumber \\&\qquad \qquad \quad \quad +\int _0^te^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)-u\cdot \nabla (m-\rho ))(\cdot ,s)ds, \end{aligned}$$

(4.3.95)

$$\begin{aligned}&\rho (\cdot ,t)=e^{t\varDelta }\rho _0-\int _0^te^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)+u\cdot \nabla \rho +\rho m)(\cdot ,s)ds, \end{aligned}$$

(4.3.96)

$$\begin{aligned}&c(\cdot ,t)=e^{t(\varDelta -1)}c_0+\int _0^te^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)ds, \end{aligned}$$

(4.3.97)

$$\begin{aligned}&u(\cdot ,t)=e^{-tA}u_0+\int _0^te^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi )(\cdot ,s)ds. \end{aligned}$$

(4.3.98)

Lemma 4.22

Under the assumptions of Proposition 4.2, we have

$$\begin{aligned} \Vert (m-\rho )(\cdot ,t)-m_\infty \Vert _{L^\theta (\varOmega )}\le M_3\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} \end{aligned}$$

for all $t\in (0,T)$ and $\theta \in [q_0,\infty ]$.

Proof

Since $e^{t\varDelta }(\overline{m}_0-\overline{\rho }_0)=m_\infty$ and $\int _\varOmega (m_0-\rho _0-m_\infty )=0$, from the Definition of T and Lemma 1.1(i), we get

$$\begin{aligned}&\Vert (m-\rho )(\cdot ,t)-m_\infty \Vert _{L^\theta (\varOmega )} \\ \le&\Vert (m-\rho )(\cdot ,t)-e^{t\varDelta }(m_0-\rho _0)\Vert _{L^\theta (\varOmega )} +\Vert e^{t\varDelta }(m_0-\rho _0)-e^{t\varDelta }m_{\infty }\Vert _{L^\theta (\varOmega )} \\ \le&\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} +c_1(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})(\Vert \rho _0\Vert _{L^{p_0}(\varOmega )} +\Vert m_0-m_\infty \Vert _{L^{p_0}(\varOmega )})e^{-\lambda _1t} \\ \le&(1+ c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} \end{aligned}$$

for all $t\in (0,T)$ and $\theta \in [q_0,\infty ]$. This lemma is proved for

$$M_3=1+ c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}.$$

Lemma 4.23

Under the assumptions of Proposition 4.2, we have

$$\begin{aligned} \Vert m(\cdot ,t)-m_\infty \Vert _{L^{\theta }(\varOmega )}\le M_4\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) e^{-\alpha _1 t} \quad \hbox {for all}\,\,t\in (0,T), \theta \in [q_0,\infty ]. \end{aligned}$$

Proof

From Lemma 4.22 and the definition of T, it follows that

$$\begin{aligned} \Vert m(\cdot ,t)-m_\infty \Vert _{L^{\theta }(\varOmega )} \le&\Vert (m-\rho -m_\infty )(\cdot ,t)\Vert _{L^{\theta }(\varOmega )}+\Vert \rho (\cdot ,t)\Vert _{L^{\theta }(\varOmega )} \\ \le&(M_3+M_1)\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1 t}. \end{aligned}$$

The lemma is proved for $M_4=M_3+M_1$.

Lemma 4.24

Under the assumptions of Proposition 4.2, there exists $M_5>0$, such that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le M_5\varepsilon (1+t^{-\frac{1}{2}+\frac{N}{2q_0}}) e^{-\alpha _2 t} \quad ~\hbox {for all }~t\in (0,T).\end{aligned}$$

Proof

For any given $\alpha _2<\lambda _1'$, we can fix $\mu \in (\alpha _2, \lambda _1')$. By (4.3.98), Lemmas 4.1, 4.2 and $\mathscr {P}(\nabla \phi )=0$, we obtain that

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\nonumber \\ \le&c_6t^{-\frac{N}{2}(\frac{1}{N}-\frac{1}{q_0})}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} +\int _0^t\Vert e^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&c_6t^{-\frac{N}{2}(\frac{1}{N}-\frac{1}{q_0})}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} \\&+c_6c_9\int _0^te^{-\mu (t-s)}\Vert (\rho +m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_6 t^{-\frac{1}{2}+\frac{N}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} \nonumber \\&+c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\int _0^te^{-\mu (t-s)}\Vert (\rho +m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds.\nonumber \end{aligned}$$

(4.3.99)

By Lemma 4.23 and the definition of T, we get

$$\begin{aligned} \Vert (\rho +m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )} =&\Vert (m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+\Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )} \\ \le&(M_4+M_1)\varepsilon (1+s^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s}.\nonumber \end{aligned}$$

(4.3.100)

Inserting (4.3.100) into (4.3.99), and noting $\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})<1$, we have

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )} \\ \le&c_6t^{-\frac{1}{2}+\frac{N}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} \\&+c_6c_9(M_4+M_1)\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\varepsilon \int _0^t(1+s^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s}e^{-\mu (t-s)}ds \\ \le&c_6 t^{-\frac{1}{2}+\frac{N}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^N(\varOmega )} \\&+c_6c_9 c_{10}(M_4+M_1)\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\varepsilon (1+t^{\min \{0,1-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}})e^{-\alpha _2t} \\ \le&c_6 t^{-\frac{1}{2}+\frac{N}{2q_0}}\varepsilon e^{-\mu t} +2c_6c_9 c_{10}(M_4+M_1) \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\varepsilon e^{-\alpha _2 t} \\ =&M_5\varepsilon (1+t^{-\frac{1}{2}+\frac{N}{2q_0}})e^{-\alpha _2 t} \end{aligned}$$

with $M_5=c_6+2c_6c_9 c_{10}(M_4+M_1)\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}$.

Lemma 4.25

Under the assumptions of Proposition 4.2, we have

$$\begin{aligned} \Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le \frac{M_2}{2}\varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _1 t}\quad \hbox {for all}\,\, t\!\in \!(0,T). \end{aligned}$$

Proof

From (4.3.97) and Lemma 1.1(iii), we have

$$\begin{aligned}&\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\nonumber \\ \le&\Vert e^{t(\varDelta -1)}\nabla c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\ \le&c_3(1+t^{-\frac{1}{2}})e^{-(\lambda _1+1)t}\Vert \nabla c_0\Vert _{L^N(\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-m_{\infty })(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds.\nonumber \end{aligned}$$

(4.3.101)

In the second inequality, we have used $\nabla e^{(t-s)(\varDelta -1)}m_{\infty }=0$.

From Lemmas 1.1, 4.3 and 4.23, it follows that

$$\begin{aligned}&\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-m_{\infty })(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert (m-m_{\infty }) (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_2 M_4\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)} (1+s^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-\alpha _1 s}ds\nonumber \\ \le&c_2 c_{10}M_4\varepsilon (1+t^{\min \{0,\frac{1}{2}-\frac{N}{2p_0}\}})e^{-\min \{\alpha _1,\lambda _1+1\}t}\nonumber \\ \le&c_2 c_{10}M_4\varepsilon (1+t^{-\frac{1}{2}})e^{-\alpha _1t}.\nonumber \end{aligned}$$

(4.3.102)

On the other hand, by Lemmas 1.1(ii), 4.3 and the definition of T, we obtain

$$\begin{aligned}&\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u\cdot \nabla c(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c(\cdot ,s) \Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2M_5M_2\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})e^{-(\lambda _1+1)(t-s)} (1+s^{-\frac{1}{2}+\frac{N}{2q_0}}) (1+s^{-\frac{1}{2}}) e^{-(\alpha _1+\alpha _2) s} \nonumber \\ \le&3c_2M_5M_2\varepsilon ^{2}\int _0^te^{-(\lambda _1+1)(t-s)} e^{-(\alpha _1+\alpha _2) s}(1+(t-s)^{-\frac{1}{2}-\frac{N}{2q_0}})(1+s^{-1+\frac{N}{2q_0}})ds\nonumber \\ \le&3c_2M_5M_2c_{10}\varepsilon ^{2}(1+t^{-\frac{1}{2}})e^{-\min \{\lambda _1+1,\alpha _1+\alpha _2\}t}\nonumber \\ \le&3c_2M_5M_2c_{10}\varepsilon ^{2}(1+t^{-\frac{1}{2}})e^{-\alpha _1t}. \nonumber \end{aligned}$$

(4.3.103)

Hence, combining above inequalities with (4.3.87) and (4.3.88), we arrive at the conclusion.

Lemma 4.26

Under the assumptions of Proposition 4.2, we have

$$\begin{aligned} \Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} \quad \hbox {for all}\,\,t\in (0,T),\,\theta \in [q_0,\infty ]. \end{aligned}$$

Proof

By the variation-of-constants formula, we have

$$\begin{aligned} \rho (\cdot ,t)=&e^{t(\varDelta -m_\infty )}\rho _0-\int _0^te^{(t-s)(\varDelta -m_\infty )}(\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)-u\cdot \nabla \rho )(\cdot ,s)ds\\&+ \int _0^te^{(t-s)(\varDelta -m_\infty )} \rho (m_\infty -m)(\cdot ,s)ds . \end{aligned}$$

By Lemma 1.1, the result in Sect. 2 of Horstmann and Winkler (2005) and $\alpha _1<\min \{\lambda _1,m_\infty \}$, we obtain

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )} \\ \le&e^{-m_\infty t}(\Vert e^{t\varDelta }(\rho _0-\overline{\rho }_0)\Vert _{L^\theta (\varOmega )}+ \Vert \overline{\rho }_0 \Vert _{L^\theta (\varOmega )}) \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}(u\cdot \nabla \rho )(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}\rho (m_\infty -m)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&c_1(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-(\lambda _1+m_\infty )t}\Vert \rho _0-\overline{\rho }_0\Vert _{L^{p_0}(\varOmega )} \\&+(\min \{1,|\varOmega |\})^{-\frac{1}{p_0}}e^{-m_\infty t}\varepsilon \\&+c_4C_S\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })})e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds\\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty ) }\nabla \cdot (\rho u)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}\rho (m_\infty -m)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds\\ \le&(2c_1+(\min \{1,|\varOmega |\})^{-\frac{1}{p_0}}) (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) \varepsilon e^{-\alpha _1t} \\&+c_4C_S\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \\&+c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{\infty }(\varOmega )}\Vert u\Vert _{L^{q_0}(\varOmega )}ds \\&+c_1\int _0^t(1+(t-s)^{-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-m_\infty (t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert m-m_\infty \Vert _{L^{\infty }(\varOmega )}ds \\ =&(2c_1+(\min \{1,|\varOmega |\})^{-\frac{1}{p_0}}) (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) \varepsilon e^{-\alpha _1t} +I_1+I_2+I_3. \end{aligned}$$

By the definition of T, Lemmas 4.25, 4.3 and (4.3.89), we get

$$\begin{aligned} I_1&\le 3 c_4C_S M_1M_2\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}e^{-2\alpha _1 s}\nonumber \\&\quad \cdot (1+s^ {-\frac{1}{2}-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) ds\nonumber \\&\le 3c_4C_Sc_{10} M_1M_2 \varepsilon ^2 (1+t^{\min \{0,-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{\lambda _1,2\alpha _1\}t}\nonumber \\&\le \frac{M_1}{8}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}. \nonumber \end{aligned}$$

Similarly, by (4.3.91) and (4.3.92), we can also get

$$\begin{aligned} I_2&\le 3 c_4 M_1M_5\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}e^{-2\alpha _1 s}(1+s^ {-\frac{1}{2}-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) ds\nonumber \\&\le 3c_4c_{10}M_5M_1\varepsilon ^2(1+t^{\min \{0,-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{\lambda _1,2\alpha _1\}t}\nonumber \\&\le \frac{M_1}{8}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t},\nonumber \end{aligned}$$

$$\begin{aligned} I_3&\le 3 c_4 M_1M_4\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-m_\infty (t-s)} e^{-2\alpha _1 s}(1+s^{-\frac{N}{p_0}+\frac{ N}{2 q_0}}) ds\nonumber \\&\le 3c_4c_{10}M_1M_4 \varepsilon ^2(1+t^{\min \{0,-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{m_\infty ,2\alpha _1\}t}\nonumber \\&\le \frac{M_1}{8}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t},\nonumber \end{aligned}$$

respectively, where the fact that $q_0\in (N,\frac{Np_0}{2(N-p_0)})$ warrants $-\frac{N}{p_0}+\frac{ N}{2 q_0}>-1$ is used. Hence, the combination of the above inequalities yields

$$\Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}.$$

Lemma 4.27

Under the assumptions of Proposition 4.2, we have

$$\begin{aligned} \Vert (m-\rho )(\cdot ,t)-e^{t\varDelta }(m_0-\rho _0)\Vert _{L^\theta (\varOmega )}\le \frac{\varepsilon }{2}(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) e^{-\alpha _1 t} \end{aligned}$$

for $\theta \in [q_0,\infty ]$, $t\in (0,T)$.

Proof

From (4.3.95) and Lemma 1.1(iv), it follows that

$$\begin{aligned}&\Vert (m-\rho )(\cdot ,t)-e^{t\varDelta }(m_0-\rho _0)\Vert _{L^\theta (\varOmega )} \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)-u\cdot \nabla (m-\rho ))(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot ((m-\rho -m_\infty )u)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&c_4C_S \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\&+c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert u(m-\rho -m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ =&I_1+I_2. \end{aligned}$$

From the definition of T and (4.3.93), we have

$$\begin{aligned} I_1\le&c_4C_S M_1 M_2 \varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)} (1+s^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-2\alpha _1s}ds \nonumber \\&\le 3c_4C_Sc_{10} M_1M_2 \varepsilon ^2 (1+t^{\min \{0,-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{\lambda _1,2\alpha _1\}t}\nonumber \\&\le \frac{\varepsilon }{4}(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}.\nonumber \end{aligned}$$

On the other hand, from Lemmas 4.22, 4.24 and (4.3.94), it follows that

$$\begin{aligned} I_2=&c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert m-\rho -m_\infty \Vert _{L^\infty (\varOmega )}\Vert u\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&2c_4M_3 M_5\varepsilon ^{2}\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\nonumber \\&\cdot (1+s^{-\frac{N}{2p_0}})e^{-\alpha _1s} (1+s^{-\frac{1}{2}+\frac{N}{2q_0}})e^{-\alpha _2 s}ds\nonumber \\ \le&6c_4M_3 M_5\varepsilon ^{2}\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) (1+s^{-\frac{1}{2}+\frac{N}{2}(\frac{1}{q_0}-\frac{1}{p_0})})\nonumber \\&\cdot e^{-\lambda _1(t-s)}e^{-(\alpha _1+\alpha _2) s}ds\nonumber \\ \le&6c_{10}c_4M_3 M_5\varepsilon ^{2} e^{-\min \{\lambda _1, \alpha _1+\alpha _2\}t}(1+t^{\min \{0,\frac{N}{2} (\frac{1}{\theta }-\frac{1}{p_0})\}})\nonumber \\ \le&\frac{\varepsilon }{4}(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1 t}.\nonumber \end{aligned}$$

Combining the above inequalities, we arrive at

$$ \Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )}\le \frac{\varepsilon }{2}(1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta } )}) e^{-\alpha _1 t}, $$

and thus complete the proof of this lemma.

By the above lemmas, we can claim that $T=T_{max}$. Indeed, if $T<T_{max}$, by Lemmas 4.27, 4.26 and 4.25, we have

$$ \Vert (m-\rho )(\cdot ,t)-e^{t\varDelta }(m_0-\rho _0)\Vert _{L^\theta (\varOmega )}\le \frac{\varepsilon }{2} (1+t^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{\theta }}) e^{-\alpha _1 t}, $$

$$ \Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{N}{2}\left( \frac{1}{p_0}-\frac{1}{\theta }\right) })e^{-\alpha _1t} $$

as well as

$$ \Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le \frac{M_2}{2}\varepsilon \left( 1+t^{-\frac{1}{2}}\right) e^{-\alpha _1 t} $$

for all $\theta \in [q_0,\infty ]$ and $t\in (0,T)$, which contradict the definition of T in (4.3.94). Next, the further estimates of solutions are established to ensure $T_{max}=\infty$.

Lemma 4.28

Under the assumptions of Proposition 4.2, there exists $M_6>0$ such that

$$\begin{aligned} \Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}\le \varepsilon M_6e^{-\alpha _2 t} \quad \hbox {for}\,\, t\in (t_0,T_{max})\,\hbox { with}\, \,t_0=\min \{\frac{T_{max}}{6},1\}.\end{aligned}$$

Proof

For any given $\alpha _2<\lambda _1'$, we can fix $\mu \in (\alpha _2, \lambda _1')$. From (4.3.98), it follows that

$$\begin{aligned}&\Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )} \\ \le&\Vert A^\beta e^{-tA} u_0 \Vert _{L^2(\varOmega )}+\int _0^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((\rho +m-m_\infty )\nabla \phi )(\cdot ,s)\Vert _{L^2(\varOmega )}ds. \end{aligned}$$

In the first integral, we apply Lemma 4.1, which gives

$$ \Vert A^\beta e^{-tA} u_0\Vert _{L^2(\varOmega )}\le c_5|\varOmega |^{\frac{N-2}{2N}} t^{-\beta }e^{-\alpha _2 t}\Vert u_0\Vert _{L^N(\varOmega )} \le c_5 |\varOmega |^{\frac{N-2}{2N}} t^{-\beta }e^{-\alpha _2 t}\varepsilon $$

for all $t\in (0,T)$. Next by Lemmas 4.2, 4.22 and 4.26, we have

$$\begin{aligned}&\int _0^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((\rho +m-m_\infty )\nabla \phi )(\cdot ,s)\Vert _{L^2(\varOmega )}ds \\ \le&c_9c_5\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} \int _0^t e^{-\mu (t-s)}(t-s)^{-\beta } \\&\cdot (\Vert m(\cdot ,s)-\rho (\cdot ,s)-m_\infty \Vert _{L^{q_0}(\varOmega )}+2\Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )})ds \\ \le&M_6'\varepsilon \int _0^t e^{-\mu (t-s)}(t-s)^{-\beta } (1+s ^{-\frac{N}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1 s}ds \\ \le&M_6' \varepsilon c_{10} (1+t^{-1})e^{-\alpha _2 t}, \end{aligned}$$

where $M_6'=(M_3+M_1)c_9c_5\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}}$. Therefore there exists $M_6>0$ such that $\Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}\le \varepsilon M_6 e^{-\alpha _2 t}$ for $t\in (t_0,T_{max})$.

Lemma 4.29

Under the assumptions of Proposition 4.2, there exists $M_7>0$, such that $\Vert c(\cdot ,t)-m_\infty \Vert _{L^{\infty }(\varOmega )}\le M_7 e^{-\alpha _2 t}$ for all $(t_0,T_{max})$ with $t_0=\min \{\frac{T_{max}}{6},1\}$.

Proof

From (4.3.97) and Lemma 1.1, we have

$$\begin{aligned}&\Vert (c-m_\infty )(\cdot ,t)\Vert _{L^\infty (\varOmega )}\nonumber \\ \le&c_1e^{-t}\Vert c_0-m_\infty \Vert _{L^\infty (\varOmega )}+\int _0^t\Vert e^{(t-s)(\varDelta -1)}(m-m_\infty )(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&+\int _0^t\Vert e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds. \end{aligned}$$

(4.3.104)

By Lemmas 4.3 and 4.23, we obtain

$$\begin{aligned}&\int _0^t\Vert e^{(t-s)(\varDelta -1)}(m-m_\infty )(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_1 \int _0^t(1+(t-s)^{-\frac{N}{2q_0}})e^{-(t-s)}\Vert (m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_1c_{10}M_4\varepsilon e^{-\alpha _2t}.\nonumber \end{aligned}$$

(4.3.105)

On the other hand, by Lemmas 4.3, 4.24 and 4.25, we get

$$\begin{aligned}&\int _0^t\Vert e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_1 \int _0^t(1+(t-s)^{-\frac{N}{2q_0}})e^{-(t-s)}\Vert u\cdot \nabla c (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&c_1 \int _0^t(1+(t-s)^{-\frac{N}{2q_0}})e^{-(t-s)}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&6c_1M_5M_2c_{10}\varepsilon ^2 e^{-\alpha _2t}. \end{aligned}$$

(4.3.106)

Therefore combining the above equalities, we arrive at the desired result.

Proof of Theorem 4.3 in the case $\mathscr {S}=0$ on $\partial \varOmega$, part 2 (Proposition 4.2).  We now come to the final step to show that $T_{max}=\infty$. According to the extensibility criterion in Lemma 4.4, it remains to show that there exists $C>0$ such that for $t_0:=\min \{\frac{T_{max}}{6},1\}<t<T_{max}$

$$ \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}<C.$$

From Lemmas 4.23 and 4.26, there exists $K_i>0$, $i=1,2,3$, such that

$$\begin{aligned}&\Vert m(\cdot ,t)-m_\infty \Vert _{L^{\infty }(\varOmega )}\le K_1 e^{-\alpha _1 t}, \\&\Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\&\Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le K_3 e^{-\alpha _1 t} \end{aligned}$$

for $t\in (t_0,T_{max})$. Furthermore, Lemma 4.29 implies that $\Vert c(\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3'e^{-\alpha _2t}$ with some $K_3'>0$ for all $t\in (t_0,T_{max})$. Since $D(A^\beta )\hookrightarrow L^\infty (\varOmega )$ with $\beta \in (\frac{N}{4},1)$, it follows from Lemma 4.28 that $\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}$ for some $K_4>0$ for all $t\in (t_0,T_{max})$. This completes the proof of Proposition 4.2.

Before we move to the next section, we remark that the following result is also valid by suitably adjusting $\varepsilon >0$ for the larger values of $p_0$ or $q_0$.

Corollary 4.1

Let $N=3$ and $\int _{\varOmega }\rho _0\ne \int _{\varOmega }m_0$. Further, let $p_0\in (\frac{N}{2}, \infty )$, $q_0\in (N,\infty )$ if $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$, and $p_0\in (\frac{2N}{3}, \infty )$, $q_0\in (N,\infty )$ if $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$. There exists $\varepsilon >0$ such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.7) as well as

$$\Vert \rho _0-\rho _\infty \Vert _{L^{p_0}(\varOmega )}\le \varepsilon ,\quad \Vert m_0-m_\infty \Vert _{L^{q_0}(\varOmega )}\le \varepsilon , \quad \Vert \nabla c_0\Vert _{L^{N}(\varOmega )}\le \varepsilon , \quad \Vert u_0\Vert _{L^{N}(\varOmega )}\le \varepsilon ,$$

(4.1.1) admits a global classical solution $(\rho ,m,c,u,P)$. Moreover, for any $\alpha _1$ $\in (0,\min \{\lambda _1, m_\infty +\rho _\infty \})$, $\alpha _2\in (0,\min \{\alpha _1,\lambda _1',1\})$, there exist constants $K_i$ $i=1,2,3,4$, such that for all $t\ge 1$

$$\begin{aligned}&\Vert m(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t},\quad \Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\&\Vert c(\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t},\quad \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

4.3.3 Global Boundedness and Decay for General $\mathscr {S}$

In this subsection, we give the proof of our results for the general matrix-valued $\mathscr {S}$. This is accomplished by an approximation procedure. In order to make the previous results applicable, we introduce a family of smooth functions $\rho _\eta \in C_0^\infty (\varOmega )$ and $0\le \rho _\eta (x)\le 1$ for $\eta \in (0,1),$ $\lim _{\eta \rightarrow 0}\rho _\eta (x)=1$ and let $\mathscr {S}_\eta (x,\rho ,c)=\rho _\eta (x)\mathscr {S}(x,\rho ,c).$ Using this definition, we regularize (4.1.1) as follows:

$$\begin{aligned} \left\{ \begin{aligned}&(\rho _\eta )_t+u_\eta \cdot \nabla \rho _\eta =\varDelta \rho _\eta -\nabla \cdot (\rho _\eta \mathscr {S}_\eta (x,\rho _\eta ,c_\eta )\nabla c_\eta )-\rho _\eta m_\eta , \\&(m_\eta )_t+u_\eta \cdot \nabla m_\eta =\varDelta m_\eta -\rho _\eta m_\eta , \\&(c_\eta )_t+u_\eta \cdot \nabla c_\eta =\varDelta c_\eta -c_\eta +m_\eta , \\&(u_\eta )_t=\varDelta u_\eta -\nabla P_\eta +(\rho _\eta +m_\eta )\nabla \phi ,\quad \nabla \cdot u_\eta =0,\\&\displaystyle \frac{\partial \rho _\eta }{\partial \nu }=\frac{\partial m_\eta }{\partial \nu } =\frac{\partial c_\eta }{\partial \nu }=0,~ u_\eta =0 \end{aligned} \right. \end{aligned}$$

(4.3.107)

with the initial data

$$\begin{aligned} \rho _\eta (x,0)=\rho _0(x),~m_\eta (x,0)=m_0(x),~c(x,0)=c_0(x),~u_\eta (x,0)=u_0(x),~ x\in \varOmega . \end{aligned}$$

(4.3.108)

It is observed that $\mathscr {S}_\eta$ satisfies the additional condition $\mathscr {S}=0$ on $\partial \varOmega$. Therefore, based on the discussion in Sect. 4.3.2, under the assumptions of Theorem 4.1 and Theorem 4.3, the problem (4.3.107)–(4.3.108) admits a global classical solution $(\rho _\eta ,m_\eta ,c_\eta ,u_\eta , P_\eta )$ that satisfies

$$\begin{aligned} \Vert m_\eta (\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t},\quad \Vert \rho _\eta (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\ \Vert c_\eta (\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \quad \Vert u_\eta (\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

for some constants $K_i$, $i=1,2,3,4$, and $t\ge 0$. Applying a standard procedure such as in Lemmas 5.2 and 5.6 of Cao and Lankeit (2016), one can obtain a subsequence of $\{\eta _j\}_{j\in \mathbb {N}}$ with $\eta _j\rightarrow 0$ as $j\rightarrow \infty$, such that $\rho _{\eta _j}\rightarrow \rho , ~m_{\eta _j}\rightarrow m, ~c_{\eta _j}\rightarrow c, u_{\eta _j}\rightarrow u \quad \hbox {in}~ C_{loc}^{\vartheta ,\frac{\vartheta }{2}}(\overline{\varOmega }\times (0,\infty ))$ as $j\rightarrow \infty$ for some $\vartheta \in (0,1)$. Moreover, by the arguments as in Lemmas 5.7, 5.8 of Cao and Lankeit (2016), one can also show that $(\rho ,m,c,u, P)$ is a classical solution of (4.1.1) with the decay properties asserted in Theorems 4.2 and 4.3. The proofs of Theorems 4.1–4.3 are thus complete.

4.4 Asymptotic Behavior of Solutions to a Coral Fertilization Model

4.4.1 A Convenient Extensibility Criterion

Firstly, we recall the result of the local existence of classical solutions, which can be proved by a straightforward adaptation of a well-known fixed point argument (see Winkler (2012) for example).

Lemma 4.30

Suppose that (4.1.14), (4.1.15) and

$$\begin{aligned} \mathscr {S}(x,\rho ,c)=0, ~~(x,\rho ,c)\in \partial \varOmega \times [0,\infty )\times [0,\infty ) \end{aligned}$$

(4.4.1)

hold. Then there exist $T_{max}\in (0,\infty ]$ and a classical solution $(\rho ,m,c,u,P)$ of (4.1.13) on $(0,T_{max})$. Moreover, $\rho ,m,c$ are nonnegative in $\varOmega \times (0,T_{max})$, and if $T_{max}<\infty$, then for $\beta \in (\frac{3}{4},1)$, as $t\rightarrow T_{max}$

$$\Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}\rightarrow \infty . $$

This solution is unique, up to addition of constants to P.

The following elementary properties of the solutions in Lemma 4.30 are immediate consequences of the integration of the first and second equations in (4.1.13), as well as an application of the maximum principle to the second and third equations.

Lemma 4.31

Suppose that (4.1.14), (4.1.15) and (4.4.1) hold. Then for all $t\in (0,T_{max})$, the solution of (4.1.13) from Lemma 4.30 satisfies

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^1(\varOmega )}\le \Vert \rho _0\Vert _{L^1(\varOmega )},\quad \Vert m(\cdot ,t)\Vert _{L^1(\varOmega )}\le \Vert m_0\Vert _{L^1(\varOmega )}, \end{aligned}$$

(4.4.2)

$$\begin{aligned}&\int _0^t\Vert \rho (\cdot ,s)m(\cdot ,s)\Vert _{L^1(\varOmega )}ds\le \min \{\Vert \rho _0\Vert _{L^1(\varOmega )},\Vert m_0\Vert _{L^1(\varOmega )}\}, \end{aligned}$$

(4.4.3)

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^1(\varOmega )}-\Vert m(\cdot ,t)\Vert _{L^1(\varOmega )}=\Vert \rho _0\Vert _{L^1(\varOmega )}-\Vert m_0\Vert _{L^1(\varOmega )}, \end{aligned}$$

(4.4.4)

$$\begin{aligned}&\Vert m(\cdot ,t)\Vert _{L^2(\varOmega )}^2+2\int _0^t\Vert \nabla m(\cdot ,s)\Vert _{L^2(\varOmega )}^2ds\le \Vert m_0\Vert _{L^2(\varOmega )}^2, \end{aligned}$$

(4.4.5)

$$\begin{aligned}&\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \Vert m_0\Vert _{L^\infty (\varOmega )}, \end{aligned}$$

(4.4.6)

$$\begin{aligned}&\Vert c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \max \{\Vert m_0\Vert _{L^\infty (\varOmega )},\Vert c_0\Vert _{L^\infty (\varOmega )}\}. \end{aligned}$$

(4.4.7)

4.4.2 Global Boundedness and Decay for $\mathscr {S}=0$ on $\partial \varOmega$

Throughout this section, we assume that $\mathscr {S}=0$ on $\partial \varOmega$. We note that, under this assumption, the boundary condition for $\rho$ in (4.1.13) reduces to the homogeneous Neumann condition $\nabla \rho \cdot \nu =0$.

In the case $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$, i.e., $\rho _\infty >0$, $m_\infty =0$, Theorem 4.4 reduces to:

Proposition 4.3

Suppose that (4.1.14) hold and $\int _{\varOmega }\rho _0>\int _{\varOmega }m_0$. Let $p_0\in (\frac{3}{2}, 3)$, $q_0\in (3,\frac{3p_0}{3-p_0})$. There exists $\varepsilon >0$, such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.15) as well as

$$\begin{aligned} \Vert \rho _0-\rho _\infty \Vert _{L^{p_0}(\varOmega )}<\varepsilon ,\quad \Vert m_0\Vert _{L^{q_0}(\varOmega )}<\varepsilon , \quad \Vert c_0\Vert _{L^{\infty }(\varOmega )}<\varepsilon , \quad \Vert u_0\Vert _{L^{3}(\varOmega )}<\varepsilon , \end{aligned}$$

(4.1.13) admits a global classical solution $(\rho ,m,c,u,P)$. In particular, for any $\alpha _1\in (0,\min \{\lambda _1,\rho _\infty \})$, $\alpha _2\in (0,\min \{\alpha _1,\lambda _1',1\})$, there exist constants $K_i$, $i=1,2,3,4$, such that for all $t\ge 1$

$$\begin{aligned}&\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t}, \end{aligned}$$

(4.4.8)

$$\begin{aligned}&\Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \end{aligned}$$

(4.4.9)

$$\begin{aligned}&\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \end{aligned}$$

(4.4.10)

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

(4.4.11)

Proposition 4.3 is the consequence of the following lemmas. In the proofs thereof, the constants $c_i$, $i=1,2,3,4$ refer to those in Lemma 1.1, $c_i>0$, $i=5,\ldots ,10$, refer to those in Lemmas 4.1–4.3.

Lemma 4.32

Under the assumptions of Proposition 4.3 and

$$\sigma =\int _0^\infty \left( 1+s^{-\frac{3}{2p_0}}\right) e^{-\alpha _1s}ds,$$

there exist $M_1>0,M_2>0$ and $\varepsilon \in (0,1)$, such that

$$\begin{aligned}&c_2+2 c_2c_{10} e^{(1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma } \le \displaystyle \frac{M_2}{4}, \end{aligned}$$

(4.4.12)

$$\begin{aligned}&c_4 c_{10} C_SM_2 (e^{(1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma }+\rho _\infty |\varOmega |^{\frac{1}{q_0}})\le \displaystyle \frac{M_1}{8}, \end{aligned}$$

(4.4.13)

$$\begin{aligned}&c_6+ 2c_6c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}(M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}\!-\!\frac{1}{q_0}}+4e^{(1+c_1+ c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma })<\frac{M_3}{4}, \end{aligned}$$

(4.4.14)

$$\begin{aligned}&c_7+ 2c_7c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{1}{3}-\frac{1}{q_0}}(M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+4e^{(1+c_1+ c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma })\nonumber \\&\;<\frac{M_4}{4}, \end{aligned}$$

(4.4.15)

$$\begin{aligned}&3 c_{10} c_4C_S(M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})M_2\varepsilon \le \displaystyle \frac{M_1}{8}, \end{aligned}$$

(4.4.16)

$$\begin{aligned}&3 c_{10}c_4 (M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})M_3 \varepsilon \le \frac{M_1}{4}, \end{aligned}$$

(4.4.17)

$$\begin{aligned}&12 c_2 c_{10} M_3 \varepsilon <1,\quad 12 c_7c_9 c_{10}M_3 \varepsilon \le 1, \quad 12 c_6c_9 c_{10}M_4 \varepsilon \le 1 . \end{aligned}$$

(4.4.18)

Let

$$\begin{aligned} \!\!T\!\triangleq \!\sup \!\left\{ \!\widetilde{T}\!\in \!(0,T_{max})\!\left| \begin{aligned}&\!\Vert (\rho \!-\!m)(\cdot ,t)\!-\!e^{t\varDelta }(\rho _0\!-\!m_0)\Vert _{L^{\theta }(\varOmega )}\! \\&\le \! M_1\varepsilon (1\!+\!t^{-\frac{3}{2} (\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}\quad \forall \theta \in [q_0,\infty ], ~t\in [0,\widetilde{T}); \\&\!\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le M_2\varepsilon (1+t^{-\frac{1}{2}})e^{-\alpha _1t}\quad \forall ~t\in [0,\widetilde{T}); \\&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le M_3\varepsilon (1+t^{-\frac{1}{2}+\frac{3}{2q_0}}) e^{-\alpha _2 t} \quad \forall t\in [0,\widetilde{T}); \\&\Vert \nabla u(\cdot ,t)\Vert _{L^{3}(\varOmega )}\le M_4\varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _2 t} \quad \forall t\in [0,\widetilde{T}). \end{aligned} \!\!\right. \right\} . \end{aligned}$$

(4.4.19)

Then $T>0$ is well-defined by Lemma 4.30 and (4.1.15). Now we claim that $T=T_{max}=\infty$ if $\varepsilon$ is sufficiently small. To this end, by the contradiction argument, it suffices to verify that all of the estimates mentioned in (4.4.19) still hold for even smaller coefficients on the right-hand side. This mainly relies on $L^p-L^q$ estimates for the Neumann heat semigroup and the fact that the classical solution on $(0,T_{max})$ can be written as

$$\begin{aligned}&(\rho -m)(\cdot ,t)\nonumber \\ =&e^{t\varDelta }(\rho _0-m_0)-\!\int _0^te^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)+ u\cdot \nabla (\rho -m))(\cdot ,s)ds, \end{aligned}$$

(4.4.20)

$$\begin{aligned} m(\cdot ,t)=&e^{t\varDelta }m_0-\int _0^te^{(t-s)\varDelta }(\rho m+u\cdot \nabla m)(\cdot ,s)ds, \end{aligned}$$

(4.4.21)

$$\begin{aligned} c(\cdot ,t)=&e^{t(\varDelta -1)}c_0+\int _0^te^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)ds, \end{aligned}$$

(4.4.22)

$$\begin{aligned} u(\cdot ,t)=&e^{-tA}u_0+\int _0^te^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi - (u\cdot \nabla )u )(\cdot ,s)ds \end{aligned}$$

(4.4.23)

for all $t\in (0,T_{max})$ according to the variation-of-constants formula.

Although the proofs of Lemmas 4.33 and 4.34 below are similar to those of Lemmas 3.11 and 3.12 in Li et al. (2019b), respectively, we provide their proofs for the convenience of the interested reader.

Lemma 4.33

Under the assumptions of Proposition 4.3, for all $t\in (0,T)$ and $\theta \in [q_0,\infty ]$, there exists constant $M_5>0$, such that

$$\begin{aligned} \Vert (\rho -m)(\cdot ,t)-\rho _\infty \Vert _{L^\theta (\varOmega )}\le M_5\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}. \end{aligned}$$

Proof

Due to $e^{t\varDelta }\rho _\infty =\rho _\infty$ and $\int _\varOmega (\rho _0-m_0-\rho _\infty )=0$, the definition of T and Lemma 1.1(i) show that for all $t\in (0,T)$ and $\theta \in [q_0,\infty ]$,

$$\begin{aligned}&\Vert (\rho -m)(\cdot ,t)-\rho _\infty \Vert _{L^\theta (\varOmega )} \\ \le&\Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )} +\Vert e^{t\varDelta }(\rho _0-m_0-\rho _\infty )\Vert _{L^\theta (\varOmega )} \\ \le&M_1\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} \\&+c_1(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})(\Vert \rho _0-\rho _\infty \Vert _{L^{p_0}(\varOmega )} +\Vert m_0\Vert _{L^{p_0}(\varOmega )})e^{-\lambda _1t} \\ \le&M_5\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}, \end{aligned}$$

where $M_5=M_1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}$.

Lemma 4.34

Under the assumptions of Proposition 4.3, for any $k>1$,

$$\begin{aligned} \Vert m(\cdot ,t)\Vert _{L^k(\varOmega )}\le M_6 \Vert m_0\Vert _{L^k(\varOmega )} e^{-\rho _\infty t}\quad \hbox {for all}~ t\in (0,T) \end{aligned}$$

(4.4.24)

with $\sigma =\int _0^\infty (1+s^{-\frac{3}{2p_0}})e^{-\alpha _1s}ds$ and $M_6=e^{M_5\sigma \varepsilon }$.

Proof

Testing the first equation in (4.1.13) with $m^{k-1}$ ($k>1$) and integrating by parts, we have

$$ \frac{d}{dt}\int _\varOmega m^k\le -k\int _\varOmega \rho m^k ~~\hbox {on}~~(0,T). $$

In view of $-\rho \le |\rho -m-\rho _\infty |-m-\rho _\infty \le -\rho _\infty +|\rho -m-\rho _\infty |$, Lemma 4.33 yields

$$\begin{aligned} \frac{d}{dt}\int _\varOmega m^k&\le -k\rho _\infty \int _\varOmega m^k+k\int _\varOmega m^k|\rho -m-\rho _\infty | \\&\le -k\rho _\infty \int _\varOmega m^k+k\Vert \rho -m-\rho _\infty \Vert _{L^\infty (\varOmega )}\int _\varOmega m^k \\&\le -k\rho _\infty \int _\varOmega m^k+kM_5\varepsilon (1+t^{-\frac{3}{2p_0}})e^{-\alpha _1t}\int _\varOmega m^k \end{aligned}$$

and thus

$$ \int _\varOmega m^k \le \int _\varOmega m_0^k \exp \{-k\rho _\infty t+kM_5\varepsilon \int _0^t(1+s^{-\frac{3}{2p_0}})e^{-\alpha _1s}ds\} \le \Vert m_0\Vert _{L^k(\varOmega )} ^ke^{k(M_5\sigma \varepsilon -\rho _\infty t)}, $$

from which (4.4.24) follows immediately.

Lemma 4.35

Under the assumptions of Proposition 4.3, we have

$$ \Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le \frac{M_3}{2} \varepsilon \left( 1+t^{-\frac{1}{2}+\frac{3}{2q_0}}\right) e^{-\alpha _2 t}~~\hbox {for all}~~t\in (0,T). $$

Proof

For $\alpha _2<\lambda _1'$, we fix $\mu \in (\alpha _2, \lambda _1')$. According to (4.4.23), Lemmas 4.1(ii) and 4.2, we infer that

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\nonumber \\ \le&c_6t^{-\frac{3}{2}\left( \frac{1}{3}-\frac{1}{q_0}\right) }e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )}\nonumber \\&+\int _0^t\Vert e^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi - (u \cdot \nabla )u)(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&c_6t^{-\frac{3}{2}\left( \frac{1}{3}-\frac{1}{q_0}\right) }e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )} \nonumber \\&+c_6\int _0^te^{-\mu (t-s)}\Vert \mathscr {P}((\rho +m-\overline{\rho +m})\nabla \phi )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&+c_6\int _0^te^{-\mu (t-s)}\Vert \mathscr {P}((u \cdot \nabla )u)(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_6 t^{-\frac{1}{2}+\frac{3}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )} \nonumber \\&+c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\int _0^te^{-\mu (t-s)}\Vert (\rho +m-\overline{\rho +m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&+c_6c_9 \int _0^t(t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}\Vert (u \cdot \nabla )u(\cdot ,s)\Vert _{L^{\frac{1}{\frac{1}{3}+\frac{1}{q_0}}}(\varOmega )}ds \nonumber \\ =:&c_6 t^{-\frac{1}{2}+\frac{3}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )}+J_1+J_2,\nonumber \end{aligned}$$

(4.4.25)

where $\mathscr {P}(\overline{\rho +m}\nabla \phi )=\overline{\rho +m} \mathscr {P}(\nabla \phi )=0$ is used.

Due to $\alpha _1<\rho _\infty$, an application of Lemmas 4.33 and 4.34 shows that

$$\begin{aligned} J_1\le&c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\int _0^te^{-\mu (t-s)}\Vert (\rho -m-\overline{\rho -m})(\cdot ,s)+2(m-\overline{m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \nonumber \\ \le&c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\int _0^te^{-\mu (t-s)}(\Vert (\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+2\Vert (m-\overline{m})(\cdot ,s)\Vert _{L^{q_0} (\varOmega )})ds \nonumber \\ \le&c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )} M_7' \varepsilon \int _0^te^{-\mu (t-s)} (1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s}ds \end{aligned}$$

(4.4.26)

with $M_7'=M_5+4e^{M_5 \sigma \varepsilon }$.

On the other hand, by the Hölder inequality and definition of T, we have

$$\begin{aligned} J_2\le&c_6c_9 \int _0^t(t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )} \Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )}ds \nonumber \\ \le&3 c_6c_9 M_3 M_4 \varepsilon ^2 \int _0^t(t-s)^{-\frac{1}{2}} e^{-\mu (t-s)} (1+s^{-1+\frac{3}{2q_0}}) e^{-2\alpha _2 s} ds. \end{aligned}$$

(4.4.27)

Now, plugging (4.4.26), (4.4.27) into (4.4.25) and applying Lemma 4.3, we end up with

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )} \\ \le&c_6 t^{-\frac{1}{2}+\frac{3}{2q_0}}e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )} +c_6c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}M_7'\varepsilon (1+t^{\min \{0,1-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}})e^{-\alpha _2t}\\&+3 c_6c_9 c_{10} M_3 M_4 \varepsilon ^2(1+t^{-\frac{1}{2}+\frac{3}{2q_0}} )e^{-\alpha _2 t} \\ \le&c_6 t^{-\frac{1}{2}+\frac{3}{2q_0}}e^{-\mu t}\varepsilon +2 c_6c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}M_7'\varepsilon e^{-\alpha _2t} \\&+3 c_6c_9 c_{10} M_3 M_4 \varepsilon ^2(1+t^{-\frac{1}{2}+\frac{3}{2q_0}} )e^{-\alpha _2 t} \\ \le&\frac{M_3}{2} \varepsilon (1+t^{-\frac{1}{2}+\frac{3}{2q_0}}) e^{-\alpha _2 t}, \end{aligned}$$

where (4.4.14), (4.4.18) and the fact that $\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})<1$ are used.

In the next lemma, we show that the estimate for the gradient is also preserved.

Lemma 4.36

Under the assumptions of Proposition 4.3, we have

$$ \Vert \nabla u(\cdot ,t)\Vert _{L^{3}(\varOmega )}\le \frac{M_4}{2} \varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _2 t}~~\hbox {for all}~~t\in (0,T). $$

Proof

According to (4.4.23), we have

$$ \nabla u(\cdot ,t)= \nabla e^{-tA}u_0+\int _0^t \nabla e^{-(t-s)A}(\mathscr {P}((\rho +m)\nabla \phi )- \mathscr {P}( (u\cdot \nabla )u ))(\cdot ,s)ds. $$

Applying Lemmas 4.1(iii), 4.2 and the Hölder inequality, we arrive at

$$\begin{aligned}&\Vert \nabla u(\cdot ,t)\Vert _{L^{3}(\varOmega )} \nonumber \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )} +\int _0^t\Vert \nabla e^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi - (u \cdot \nabla )u)(\cdot ,s)\Vert _{L^3(\varOmega )}ds \nonumber \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon +c_7\int _0^t (t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}\Vert \mathscr {P}((\rho +m-\overline{\rho +m})\nabla \phi )(\cdot ,s)\Vert _{L^{3}(\varOmega )}ds \nonumber \\&+c_7\int _0^t (t-s)^{-\frac{1}{2}-\frac{3}{2q_0}} e^{-\mu (t-s)}\Vert \mathscr {P}((u \cdot \nabla )u)(\cdot ,s)\Vert _ {L^{\frac{3q_0}{3+q_0}}(\varOmega )}ds \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon \nonumber \\&+c_7c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )} |\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}\int _0^t (t-s)^{-\frac{1}{2}} e^{-\mu (t-s)}\Vert (\rho +m-\overline{\rho +m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \nonumber \\&+c_7c_9\int _0^t (t-s)^{-\frac{1}{2}-\frac{3}{2q_0}} e^{-\mu (t-s)} \Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \nonumber \\ =:&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon +\tau _1+\tau _2, \nonumber \end{aligned}$$

(4.4.28)

where $\mathscr {P}(\overline{\rho +m}\nabla \phi )=\overline{\rho +m} \mathscr {P}(\nabla \phi )=0$ is used.

Due to $\alpha _1<\rho _\infty$, an application of Lemmas 4.33 and 4.34 shows that

$$\begin{aligned} \tau _1\le&c_7c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )} |\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}\int _0^t (t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}\nonumber \\&\Vert (\rho -m-\overline{\rho -m})(\cdot ,s)+2(m-\overline{m})(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_7c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )} |\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}} \int _0^t(t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}\nonumber \\&(\Vert (\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+2\Vert (m-\overline{m})(\cdot ,s)\Vert _{L^{q_0} (\varOmega )})ds \nonumber \\ \le&c_7c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )} |\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}M_7' \varepsilon \int _0^te^{-\mu (t-s)} (1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})})(t-s)^{-\frac{1}{2}}e^{-\alpha _1s}ds. \nonumber \end{aligned}$$

(4.4.29)

On the other hand, from the Hölder inequality and definition of T, it follows that

$$\begin{aligned} \tau _2 \le 3 c_7c_9 M_3 M_4 \varepsilon ^2 \int _0^t(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}} e^{-\mu (t-s)} (1+s^{-1+\frac{3}{2q_0}}) e^{-2\alpha _2 s} ds. \end{aligned}$$

(4.4.30)

Therefore, inserting (4.4.30), (4.4.29) into (4.4.28) and applying Lemma 4.3, we get

$$\begin{aligned}&\Vert \nabla u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )} \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon +c_7c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}M_7'\varepsilon (1+t^{\min \{0, \frac{1}{2}-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}})e^{-\alpha _2t}\\&+3 c_7c_9 c_{10} M_3 M_4 \varepsilon ^2(1+t^{-\frac{1}{2}} )e^{-\alpha _2 t} \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon \!+\!2 c_7c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}M_7' \varepsilon e^{-\alpha _2t} \\&+3 c_7c_9 c_{10} M_3 M_4 \varepsilon ^2(1+t^{-\frac{1}{2}} )e^{-\alpha _2 t} \\ \le&\frac{M_4}{2} \varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _2 t}, \end{aligned}$$

where (4.4.15), (4.4.18) and the fact that $q_0\in (3,\frac{3p_0}{3-p_0}), p_0\in (\frac{3}{2},3)$ are used.

Lemma 4.37

Under the assumptions of Proposition 4.3, we have

$$\begin{aligned} \Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le \frac{M_2}{2}\varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _1 t}~~\hbox {for all}~~t\in (0,T). \end{aligned}$$

Proof

By (4.4.22) and Lemma 1.1(ii), we have

$$\begin{aligned}&\quad \Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\nonumber \\&\le \Vert e^{t(\varDelta -1)}\nabla c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&\le c_2(1+t^{-\frac{1}{2}})e^{-(\lambda _1+1)t}\Vert c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}m(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&\quad +\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds. \end{aligned}$$

(4.4.31)

Now we estimate the last two integrals on the right-hand side of the above inequality. From Lemmas 1.1(ii), 4.3, 4.34 with $k=q_0$ and the fact that $q_0>3$, it follows that

$$\begin{aligned} \int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}m\Vert _{L^\infty (\varOmega )}ds \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert m\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&c_2 M_6\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)} e^{-\rho _\infty s}ds \nonumber \\ \le&c_2c_{10} M_6 (1+t^{\min \{0,\frac{1}{2}-\frac{3}{2q_0}\}})\varepsilon e^{-\alpha _1t} \\ \le&2 c_2c_{10} M_6 \varepsilon e^{-\alpha _1 t}.\nonumber \end{aligned}$$

(4.4.32)

On the other hand, by Lemmas 1.1(ii), 4.3, 4.35 and the definition of T, we obtain

$$\begin{aligned}&\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u\cdot \nabla c\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2 M_3M_2 \varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)}(1+s^{-\frac{1}{2}+\frac{3}{2q_0}}) (1+s^{-\frac{1}{2}}) e^{-(\alpha _1+\alpha _2) s}ds \nonumber \\ \le&3c_2M_3M_2\varepsilon ^{2}\int _0^te^{-(\lambda _1+1)(t-s)} e^{-(\alpha _1+\alpha _2) s}(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})(1+s^{-1+\frac{3}{2q_0}})ds\nonumber \\ \le&3c_2 c_{10}M_2 M_3\varepsilon ^2(1+t^{-\frac{1}{2}})e^{-\alpha _1 t}.\nonumber \end{aligned}$$

(4.4.33)

From (4.4.31)–(4.4.33), it follows that

$$\begin{aligned} \Vert \nabla c\Vert _{L^\infty (\varOmega )}&\le (c_2+2 c_2c_{10} M_6 +3c_2 c_{10}M_2 M_3\varepsilon ) (1+t^{-\frac{1}{2}})\varepsilon e^{-\alpha _1t} \\&\le \frac{M_2}{2}(1+t^{-\frac{1}{2}})\varepsilon e^{-\alpha _1 t}, \end{aligned}$$

due to the choice of $M_2, M_3$ and $\varepsilon$ in (4.4.12) and (4.4.18), and thereby completes the proof.

Lemma 4.38

Under the assumptions of Proposition 4.3, for all $\theta \in [q_0,\infty ]$ and $t\in (0,T)$,

$$\begin{aligned} \Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) e^{-\alpha _1 t}. \end{aligned}$$

Proof

According to (4.4.20), Lemma 1.1(iv), we have

$$\begin{aligned}&\Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )} \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)+u\cdot \nabla (\rho -m))(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot ((\rho -m-\rho _\infty )u)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&c_4C_S\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\&+c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert u(\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ =:&I_1+I_2. \end{aligned}$$

Now we need to estimate $I_1$ and $I_2$. Firstly, from Lemmas 4.33 and 4.34, we obtain

$$\begin{aligned} \Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}&\le \Vert (\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+\Vert m(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+\Vert \rho _\infty \Vert _{L^{q_0}(\varOmega )}\nonumber \\&\le M_5\varepsilon (1+s^{-\frac{3}{2}\left( \frac{1}{p_0}-\frac{1}{q_0}\right) })e^{-\alpha _1s}+M_8 \end{aligned}$$

(4.4.34)

with $M_8=e^{(1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}})\sigma }+\rho _\infty |\varOmega |^{\frac{1}{q_0}}$, which along with Lemmas 4.37 and 1.1 implies that

$$\begin{aligned} I_1\le&c_4C_SM_8\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \nonumber \\&+ c_4C_SM_5\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })})(1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-\alpha _1s} e^{-\lambda _1(t-s)}\nonumber \\&\cdot \Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \\ \le&c_4C_SM_8M_2\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}(1+s^{-\frac{1}{2}}) e^{-\alpha _1 s}ds \nonumber \\&+3 c_4C_S M_5 M_2\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })})(1+s^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-2\alpha _1s}\nonumber \\&\cdot e^{-\lambda _1(t-s)}ds\nonumber \\ \le&c_{10}c_4C_S (M_8M_2+3 M_5M_2\varepsilon )(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})\varepsilon e^{-\alpha _1 t}\nonumber \\ \le&\frac{M_1}{4}(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})\varepsilon e^{-\alpha _1 t},\nonumber \end{aligned}$$

(4.4.35)

where we have used (4.4.13) and (4.4.16) and $\frac{1}{p_0}-\frac{1}{q_0}<\frac{1}{3}$.

On the other hand, from Lemmas 4.33 and 4.35, it follows that

$$\begin{aligned} I_2&=c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\alpha _1(t-s)}\Vert \rho -m-\rho _\infty \Vert _{L^\infty (\varOmega )}\Vert u\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&\le 3c_4M_3M_5\varepsilon ^{2}\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\alpha _1(t-s)}(1+s^{-\frac{1}{2}+\frac{3 }{ 2q_0}-\frac{3}{2p_0}})\nonumber \\&\quad \cdot e^{-(\alpha _1+\alpha _2)s} ds \\&\le 3c_4c_{10}M_3M_5 \varepsilon ^{2}(1+t^{\min \{0,\frac{3}{2} (\frac{1}{\theta }-\frac{1}{p_0})\}})e^{-\alpha _1 t }\nonumber \\&\le \frac{M_1}{4}\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1 t},\nonumber \end{aligned}$$

(4.4.36)

where we have used (4.4.17) and $\frac{1}{p_0}-\frac{1}{q_0}<\frac{1}{3}$. Hence, combining the above inequalities leads to our conclusion immediately.

Now we are ready to complete the proof of Proposition 4.3.

Proof of Proposition 4.3. First from Lemmas 4.35–4.38 and Definition (4.4.19), it follows that $T=T_{max}$. It remains to show that $T_{max}=\infty$ and to establish convergence result asserted in Proposition 4.3.

Supposed that $T_{max}<\infty$. We only need to show that for all $t \le T_{max}$,

$$ \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}< \infty $$

with $\beta \in (\frac{3}{4},1)$ according to the extensibility criterion in Lemma 4.30.

Let $t_0:=\min \{1,\frac{T_{max}}{3}\}$. Then from Lemma 4.34, there exists $K_1>0$, such that for $t\in (t_0,T_{max})$,

$$\begin{aligned} \Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_1e^{-\rho _\infty t}. \end{aligned}$$

(4.4.37)

Moreover, from Lemma 4.33 and the fact that

$$ \Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le \Vert (\rho -m)(\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )}, $$

it follows that for all $t\in (t_0,T_{max})$ and some constant $K_2>0$,

$$\begin{aligned} \Vert \rho (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}. \end{aligned}$$

(4.4.38)

Furthermore, Lemma 4.37 implies that there exists $K_3'>0$, such that

$$\begin{aligned} \Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le K_3'e^{-\alpha _1 t}\quad \hbox {for all}\,\,t\in (t_0,T_{max}) \end{aligned}$$

(4.4.39)

Hence, it only remains to show that

$$ \Vert c(\cdot ,t)\Vert _{L^\infty (\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}\le C~~\hbox {for all}~t\in (t_0,T_{max}). $$

for some constant $C>0$. In fact, we will show that

$$\begin{aligned} \Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}\le C e^{-\alpha _2 t} \end{aligned}$$

(4.4.40)

for $t_0<t<T_{max}$ with some constant $C>0$.

By (4.4.23), we have

$$\begin{aligned}&\Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )} \\ \le&\Vert A^\beta e^{-tA} u_0\Vert _{L^2(\varOmega )} +\int _{0}^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((\rho +m-\rho _\infty )\nabla \phi )(\cdot ,s)\Vert _{L^2(\varOmega )}ds\nonumber \\&+\int _{0}^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((u\cdot \nabla )u )(\cdot ,s)\Vert _{L^2(\varOmega )}ds\nonumber . \end{aligned}$$

(4.4.41)

According to Lemma 4.1,

$$ \Vert A^\beta e^{-tA} u_0\Vert _{L^2(\varOmega )}\le c_5 e^{-\mu t}\Vert A^\beta u_0\Vert _{L^2(\varOmega )}~~\hbox {for all}~~t\in (0,T_{max}). $$

From Lemmas 4.1, 4.2, 4.33 and the Hölder inequality, it follows that there exists $l_1>0$, such that

$$\begin{aligned}&\int _0^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((\rho +m-\rho _\infty )\nabla \phi )(\cdot ,s)\Vert _{L^2(\varOmega )}ds \\ \le&c_5 c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} \int _0^t (\Vert (\rho -m-\rho _\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )} \\&+2\Vert m(\cdot ,s)\Vert _{L^{q_0}(\varOmega )})(t-s)^{-\beta }e^{-\mu (t-s)}ds \\ \le&c_5 c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}}l_1 \int _0^t e^{-\mu (t-s)}(t-s)^{-\beta } (1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s} ds\\ \le&c_5 c_9 c_{10}\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} l_1 e^{-\alpha _2 t}(1+t^{\min \{0,1-\beta -\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}}). \end{aligned}$$

On the other hand, let $M(t):=e^{-\alpha _2 t}\Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}$ for $0<t<T_{max}$. By Lemmas 4.1(iv) and the Gagliardo–Nirenberg type inequality, one can see that

$$\begin{aligned} \Vert (u\cdot \nabla )u(\cdot ,s)\Vert _{L^{2}(\varOmega )}\le&|\varOmega |^{\frac{1}{6}}\Vert u(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}\Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )}\\ \le&l_2 \Vert A^\beta u(\cdot ,s)\Vert _{L^2(\varOmega )}^{\vartheta }\Vert u(\cdot ,s)\Vert ^{1-\vartheta }_{L^{q_0}(\varOmega )} \Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )} \end{aligned}$$

for some $l_2>0$ with $\vartheta =\frac{1}{q_0}/(\frac{1}{q_0}-\frac{1}{2}+\frac{2\beta }{3})$, and thereby an application of Lemmas 2.2, 4.2, 4.35 and 4.36 gives

$$\begin{aligned}&\int _0^t\Vert A^\beta e^{-(t-s)A}\mathscr {P}((u\cdot \nabla )u)(\cdot ,s)\Vert _{L^2(\varOmega )}ds \\ \le&c_5 c_9 l_2 \int _0^t\Vert A^\beta u(\cdot ,s)\Vert _{L^2(\varOmega )}^{\vartheta }\Vert u(\cdot ,s)\Vert ^{1-\vartheta }_{L^{q_0}(\varOmega )} \Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )} \\ \le&l_3 (\max _{0\le s<T_{max}}M(s))^\vartheta \int _0^t e^{-\mu (t-s)}(t-s)^{-\beta } (1+s^{-\frac{1}{2}+(-\frac{1}{2}+\frac{3}{2q_0})(1-\vartheta )})e^{-2\alpha _2s} ds\\ \le&c_{10} l_3 (\max _{0\le s<T_{max}}M(s))^\vartheta (1+t^{\min \{0,\frac{1}{2}-\beta +(\frac{3}{2q_0}-\frac{1}{2})(1-\vartheta )\}})e^{-\alpha _2 t} \end{aligned}$$

for some $l_3>0$. Now inserting the above inequalities into (4.4.41), we arrive at

$$\begin{aligned} M(t)\le&c_5 \Vert A^\beta u_0\Vert _{L^2(\varOmega )}+ c_5 c_9 c_{10}\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{q_0-2}{2q_0}} l_1 (1+t^{\min \{0,1-\beta -\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}})\\&+ c_{10} l_3 (\max _{0\le s<T_{max}}M(s))^\vartheta (1+t^{\min \{0,\frac{1}{2}-\beta +(\frac{3}{2q_0}-\frac{1}{2})(1-\vartheta )\}}), \end{aligned}$$

which implies that for some $l_4>0$ depending on $t_0$, we have

$$\begin{aligned} \max _{t_0\le t<T_{max}}M(t)\le l_4+l_4 (\max _{0\le t<T_{max}}M(t))^\vartheta . \end{aligned}$$

On the other hand, from Lemma 4.30, $\displaystyle \max _{0\le t\le t_0}M(t)\le l_5.$ Therefore, we get

$$\begin{aligned} \max _{0\le t<T_{max}}M(t)\le l_4+l_5+l_4 (\max _{0\le t<T_{max}}M(t))^\vartheta . \end{aligned}$$

As $\vartheta <1$, we infer that $M(t)\le l_6$ for all $t\in (0,T_{max})$ for some $l_6>0$ independent of $T_{max}$ and hence arrive at (4.4.40).

Furthermore, due to $D(A^\beta )\hookrightarrow L^\infty (\varOmega )$ with $\beta \in (\frac{3}{4},1)$ and Lemma 4.35, we get

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t} \quad \hbox {for some}\,K_4>0 \,\hbox {and }\, t\in (0,T_{max}). \end{aligned}$$

(4.4.42)

Now we turn to showing that there exists $K_3''>0$, such that

$$\begin{aligned} \Vert c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le K_3'' e^{-\alpha _2t}\quad \hbox {for all}\,\, t\in (0,T_{max}). \end{aligned}$$

(4.4.43)

From (4.4.22), it follows that

$$\begin{aligned} \Vert c\Vert _{L^\infty (\varOmega )}&\le \Vert e^{t(\varDelta -1)}c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert e^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&\le e^{-t}\Vert c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert e^{(t-s)(\varDelta -1)}m(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\&\quad +\int _0^t\Vert e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds.\nonumber \end{aligned}$$

(4.4.44)

An application of (4.4.24) with $k=\infty$ yields

$$\begin{aligned} \int _0^t\Vert e^{(t-s)(\varDelta -1)}m(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds&\le \int _0^t e^{-(t-s)}\Vert m(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}ds \\&\le \Vert m_0\Vert _{L^\infty (\varOmega )}M_6\int _0^t e^{-(t-s)} e^{-\rho _\infty s} ds\nonumber \\&\le \Vert m_0\Vert _{L^\infty (\varOmega )}M_6c_{10} e^{-\alpha _2t}.\nonumber \end{aligned}$$

(4.4.45)

On the other hand, from (4.4.42) and (4.4.39), we can see that

$$\begin{aligned} \int _0^t\Vert e^{(t-s)(\varDelta -1)} u\cdot \nabla c\Vert _{L^\infty (\varOmega )}ds&\le \int _0^t e^{-(t-s)}\Vert u\Vert _{L^{\infty }(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \\&\le K_3' K_4 \int _0^t e^{-(\alpha _1+\alpha _2) s} e^{-(t-s)}ds \nonumber \\&\le K_3' K_4 c_{10} e^{-\alpha _2 t}. \nonumber \end{aligned}$$

(4.4.46)

Inserting (4.4.45), (4.4.46) into (4.4.44), we arrive at the conclusion (4.4.43). We have thus established that $T_{max}=\infty$, and the decay estimates in (4.4.8)–(4.4.11) follow from (4.4.37)–(4.4.40) and (4.4.43), respectively.

As for the case $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$, i.e., $m_\infty >0$, $\rho _\infty =0$, Theorem 4.5 reduces to

Proposition 4.4

Assume that (4.1.14) and $\int _{\varOmega }\rho _0<\int _{\varOmega }m_0$ hold, and let $p_0\in (2,3)$, $q_0\in (3,\frac{3p_0}{2(3-p_0)})$. Then there exists $\varepsilon >0$, such that for any initial data $(\rho _0,m_0,c_0,u_0)$ fulfilling (4.1.15) as well as

$$ \Vert \rho _0\Vert _{L^{p_0}(\varOmega )}\le \varepsilon ,\quad \Vert m_0-m_\infty \Vert _{L^{q_0}(\varOmega )}\le \varepsilon , \quad \Vert \nabla c_0\Vert _{L^{3}(\varOmega )}\le \varepsilon , \quad \Vert u_0\Vert _{L^{3}(\varOmega )}\le \varepsilon , $$

(4.1.13) admits a global classical solution $(\rho ,m,c,u,P)$. Furthermore, for any $\alpha _1\!\in \!(0,\min \{\lambda _1,m_\infty ,1\})$, $\alpha _2\!\in \!(0,\min \{\alpha _1,\lambda _1'\})$, there exist constants $K_i>0$, $i=1,2,3,4$, such that

$$\begin{aligned}&\Vert m(\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t}, \end{aligned}$$

(4.4.47)

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \end{aligned}$$

(4.4.48)

$$\begin{aligned}&\Vert c(\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \end{aligned}$$

(4.4.49)

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}. \end{aligned}$$

(4.4.50)

The basic strategy of the proof of Proposition 4.4 parallels that of Proposition 4.3 to a certain extent. However, due to differences in the properties of $\rho$ and m, there are significant differences in the details of their proofs. Thus, for the convenience of the reader, we will sketch the proof of Proposition 4.4.

The following elementary observations can be verified easily:

Lemma 4.39

Under the assumptions of Proposition 4.4, it is possible to choose $M_1>0,M_2>0$ and $\varepsilon >0$, such that

$$\begin{aligned}&c_3+ c_2 c_{10}(1+ c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+M_1)\le \displaystyle \frac{M_2}{4}, \end{aligned}$$

(4.4.51)

$$\begin{aligned}&c_6 +2c_6c_9 c_{10} (M_1\!+2+2c_1\!+2 c_1|\varOmega |^{\frac{1}{p_0}\!-\!\frac{1}{q_0}}\!) \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}< \displaystyle \frac{M_3}{4} \end{aligned}$$

(4.4.52)

$$\begin{aligned}&c_7 +2c_7c_9 c_{10} (M_1\!+2+2c_1\!+ 2c_1|\varOmega |^{\frac{1}{p_0}\!-\!\frac{1}{q_0}}\!) \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}< \displaystyle \frac{M_4}{4} \end{aligned}$$

(4.4.53)

$$\begin{aligned}&12c_2c_{10}M_3\varepsilon \le 1, \end{aligned}$$

(4.4.54)

$$\begin{aligned}&2c_1+(\min \{1,|\varOmega |\})^{-\frac{1}{p_0}}\le \displaystyle \frac{M_1}{8},\quad 12c_6c_9c_{10}M_4 \varepsilon <1, \end{aligned}$$

(4.4.55)

$$\begin{aligned}&24c_4C_Sc_{10}M_2\varepsilon <1, \end{aligned}$$

(4.4.56)

$$\begin{aligned}&12c_7c_9c_{10}M_3 \varepsilon <1, \end{aligned}$$

(4.4.57)

$$\begin{aligned}&12c_4c_{10} C_S M_1M_2 \varepsilon <1, \end{aligned}$$

(4.4.58)

$$\begin{aligned}&24c_1c_{10}(1+ c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}+M_1)\varepsilon <1, \end{aligned}$$

(4.4.59)

$$\begin{aligned}&18c_4c_{10} M_3 \varepsilon < 1. \end{aligned}$$

(4.4.60)

$$\begin{aligned}&12 c_{10} c_4M_3(1\!+\! c_1\!+\!c_1|\varOmega |^{\frac{1}{p_0}\!-\!\frac{1}{q_0}}) \varepsilon \!< 1. \end{aligned}$$

(4.4.61)

Define

$$\begin{aligned} T\!:=\!\sup \!\left\{ \!\widetilde{T}\!\!\in \!(0,T_{max})\!\!\left| \begin{aligned}&\!\Vert (m\!-\!\rho )(\cdot ,t)\!-\!e^{t\varDelta }(m_0\!-\!\rho _0)\Vert _{L^{\theta }(\varOmega )}\!\le \! \varepsilon (1+t^{-\frac{ 3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}; \\&\!\Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )}\le M_1\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t},\,\forall \theta \in [q_0,\infty ]; \\&\!\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le M_2\varepsilon (1+t^{-\frac{1}{2}})e^{-\alpha _1 t} \quad \hbox {for all }\, t\in [0,\widetilde{T});\\&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le M_3\varepsilon \left( 1+t^{-\frac{1}{2}+\frac{3}{2q_0}}\right) e^{-\alpha _2 t} ~\hbox {for all}~t\in [0,\widetilde{T});\\&\Vert \nabla u(\cdot ,t)\Vert _{L^{3}(\varOmega )}\le M_4\varepsilon \left( 1+t^{-\frac{1}{2}}\right) e^{-\alpha _2 t} ~\hbox {for all}~t\in [0,\widetilde{T}). \end{aligned}\right. \!\right\} \end{aligned}$$

(4.4.62)

By Lemma 4.30 and (4.1.15), $T>0$ is well-defined. As in the proof of Proposition 4.3, we first show $T=T_{max}$, and then $T_{max}=\infty$. To this end, we will show that all of the estimates mentioned in (4.4.62) are still valid with even smaller coefficients on the right-hand side. The derivation of these estimates will mainly rely on $L^p-L^q$ estimates for the Neumann heat semigroup and the corresponding semigroup for the Stokes operator, and the fact that the classical solution of (4.1.13) on (0, T) can be represented as

$$\begin{aligned}&(m-\rho )(\cdot ,t)=e^{t\varDelta }(m_0-\rho _0)\nonumber \\&\qquad \qquad \quad +\int _0^te^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)-u\cdot \nabla (m-\rho ))(\cdot ,s)ds, \end{aligned}$$

(4.4.63)

$$\begin{aligned}&\rho (\cdot ,t)=e^{t\varDelta }\rho _0-\int _0^te^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(x,\rho ,c)\nabla c)+u\cdot \nabla \rho +\rho m)(\cdot ,s)ds, \end{aligned}$$

(4.4.64)

$$\begin{aligned}&c(\cdot ,t)=e^{t(\varDelta -1)}c_0+\int _0^te^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)ds, \end{aligned}$$

(4.4.65)

$$\begin{aligned}&u(\cdot ,t)= e^{-tA}u_0+\int _0^te^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi - (u\cdot \nabla )u )(\cdot ,s)ds. \end{aligned}$$

(4.4.66)

Lemma 4.40

(Lemma 3.17 in Li et al. (2019b)) Under the assumptions of Proposition 4.4,

$$\begin{aligned} \Vert (m-\rho )(\cdot ,t)-m_\infty \Vert _{L^\theta (\varOmega )}\le M_5\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} \end{aligned}$$

for all $t\in (0,T)$ and $\theta \in [q_0,\infty ]$ with $M_5=1+c_1+c_1|\varOmega |^{\frac{1}{p_0}-\frac{1}{q_0}}$.

Lemma 4.41

(Lemma 3.18 in Li et al. (2019b)) Under the assumptions of Proposition 4.4,

$$\begin{aligned} \Vert m(\cdot ,t)-m_\infty \Vert _{L^{\theta }(\varOmega )}\le (M_5+M_1)\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) e^{-\alpha _1 t} \end{aligned}$$

for all $t\in (0,T)$, $\theta \in [q_0,\infty ]$.

Lemma 4.42

Under the assumptions of Proposition 4.4, we have

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le \frac{ M_3}{2}\varepsilon (1+t^{-\frac{1}{2}+\frac{3}{2q_0}}) e^{-\alpha _2 t} \quad ~\hbox {for all }~t\in (0,T).\end{aligned}$$

Proof

For any given $\alpha _2<\lambda _1'$, we can fix $\mu \in (\alpha _2, \lambda _1')$. By (4.4.66), Lemmas 4.1, 4.2 and $\mathscr {P}(\nabla \phi )=0$, we obtain that

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\nonumber \\ \le&c_6t^{-\frac{3}{2}(\frac{1}{3}-\frac{1}{q_0})}e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )}\nonumber \\&+\int _0^t\Vert e^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi -(u\cdot \nabla ) u)(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_6t^{-\frac{1}{2}+\frac{3}{2q_0}}e^{-\mu t}\varepsilon +c_6c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\int _0^te^{-\mu (t-s)}\Vert (\rho +m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&+c_6c_9 \int _0^t(t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}\Vert (u \cdot \nabla )u(\cdot ,s)\Vert _{L^{\frac{1}{\frac{1}{3}+\frac{1}{q_0}}}(\varOmega )}ds. \nonumber \end{aligned}$$

(4.4.67)

By Lemma 4.41 and the definition of T, we get

$$\begin{aligned} \Vert (\rho +m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )} =&\Vert (m-m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}+\Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )} \\ \le&(2M_5+M_1)\varepsilon (1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s}.\nonumber \end{aligned}$$

(4.4.68)

Inserting (4.4.68) into (4.4.67), by the definition of T and noting that $\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})<1$, we have

$$\begin{aligned}&\Vert u(\cdot ,t)\Vert _{L^{q_0}(\varOmega )} \\ \le&c_6t^{-\frac{1}{2}+\frac{3}{2q_0}}e^{-\mu t}\varepsilon \\&+c_6c_9(2M_5+M_1)\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\varepsilon \int _0^t(1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s}e^{-\mu (t-s)}ds \\&+c_6c_9 \int _0^t(t-s)^{-\frac{1}{2}}e^{-\mu (t-s)} \Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \nonumber \\ \le&c_6 t^{-\frac{1}{2}+\frac{3}{2q_0}}e^{-\mu t}\varepsilon +c_6c_9 c_{10}(2M_5+M_1)\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\varepsilon (1+t^{\min \{0,1-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})\}})e^{-\alpha _2t} \nonumber \\&+ 3 c_6c_9 M_3 M_4 \varepsilon ^2 \int _0^t(t-s)^{-\frac{1}{2}} e^{-\mu (t-s)} (1+s^{-1+\frac{3}{2q_0}}) e^{-2\alpha _2 s} ds \nonumber \\ \le&c_6 t^{-\frac{1}{2}+\frac{3}{2q_0}}\varepsilon e^{-\mu t} +2c_6c_9 c_{10}(2M_5+M_1) \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\varepsilon e^{-\alpha _2 t}\nonumber \\&+ 3 c_6c_9 c_{10}M_3 M_4 (1+t^{-\frac{1}{2}+\frac{3}{2q_0}})\varepsilon ^2e^{-\alpha _2 t} \nonumber \\ \le&\frac{ M_3}{2}\varepsilon (1+t^{-\frac{1}{2}+\frac{3}{2q_0}}) e^{-\alpha _2 t}, \end{aligned}$$

where we have used (4.4.52) and (4.4.55).

Lemma 4.43

Under the assumptions of Proposition 4.4, we have

$$ \Vert \nabla u(\cdot ,t)\Vert _{L^{3}(\varOmega )}\le \frac{M_4}{2} \varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _2 t}~~\hbox {for all}~~t\in (0,T). $$

Proof

According to (4.4.66), and applying Lemmas 4.1(iii) and 4.2, we arrive at

$$\begin{aligned}&\Vert \nabla u(\cdot ,t)\Vert _{L^{3}(\varOmega )}\nonumber \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\Vert u_0\Vert _{L^3(\varOmega )} +\int _0^t\Vert \nabla e^{-(t-s)A}\mathscr {P}((\rho +m)\nabla \phi - (u \cdot \nabla )u)(\cdot ,s)\Vert _{L^3(\varOmega )}ds\nonumber \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon +c_7|\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}\int _0^t (t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}\Vert \mathscr {P}((\rho +m-m_{\infty })\nabla \phi )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&+c_7\int _0^t (t-s)^{-\frac{1}{2}-\frac{3}{2q_0}} e^{-\mu (t-s)}\Vert \mathscr {P}((u \cdot \nabla )u)(\cdot ,s)\Vert _ {L^{\frac{3q_0}{3+q_0}}(\varOmega ) } ds \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon \nonumber \\&+c_7c_9\Vert \nabla \phi \Vert _{L^\infty (\varOmega )} |\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}} \int _0^t (t-s)^{-\frac{1}{2}} e^{-\mu (t-s)}\Vert (\rho +m-m_{\infty })(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&+c_7c_9\int _0^t (t-s)^{-\frac{1}{2}-\frac{3}{2q_0}} e^{-\mu (t-s)} \Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds, \nonumber \end{aligned}$$

(4.4.69)

where $\mathscr {P}(m_{\infty }\nabla \phi )=m_{\infty } \mathscr {P}(\nabla \phi )=0$ is used.

From (4.4.68), it follows that

$$\begin{aligned}&\int _0^t (t-s)^{-\frac{1}{2}} e^{-\mu (t-s)}\Vert (\rho +m-m_{\infty })(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds\\ \le&(2M_5+M_1)\varepsilon \int _0^t(t-s)^{-\frac{1}{2}}e^{-\mu (t-s)}(1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})})e^{-\alpha _1s}ds. \nonumber \end{aligned}$$

(4.4.70)

In addition, an application of the Hölder inequality and definition of T shows that

$$\begin{aligned}&\int _0^t(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}} e^{-\mu (t-s)}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )} \Vert \nabla u(\cdot ,s)\Vert _{L^{3}(\varOmega )}ds \nonumber \\ \le&3 M_3 M_4 \varepsilon ^2 \int _0^t(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}} e^{-\mu (t-s)} (1+s^{-1+\frac{3}{2q_0}}) e^{-2\alpha _2 s} ds. \end{aligned}$$

(4.4.71)

Therefore, inserting (4.4.71), (4.4.70) into (4.4.69) and applying Lemma 4.3, we get

$$\begin{aligned}&\Vert \nabla u(\cdot ,t)\Vert _{L^{3}(\varOmega )} \\ \le&c_7 t^{-\frac{1}{2}}e^{-\mu t}\varepsilon \\&+c_7c_9 c_{10} \Vert \nabla \phi \Vert _{L^\infty (\varOmega )}|\varOmega |^{\frac{1}{3}\!-\!\frac{1}{q_0}}(2M_5+M_1)\varepsilon (1+t^{\min \{0,\frac{1}{2}-\frac{3}{ 2} (\frac{1}{p_0}-\frac{1}{q_0})\}})e^{-\alpha _2t}\\&+3 c_7c_9 c_{10} M_3 M_4 \varepsilon ^2(1+t^{-\frac{1}{2}} )e^{-\alpha _2 t} \\ \le&\frac{M_4}{2} \varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _2 t}, \end{aligned}$$

where (4.4.53), (4.4.57) are used.

Lemma 4.44

Under the assumptions of Proposition 4.4, we have

$$\begin{aligned} \Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le \frac{M_2}{2}\varepsilon (1+t^{-\frac{1}{2}}) e^{-\alpha _1 t}\quad \hbox {for all}\,\, t\!\in \!(0,T). \end{aligned}$$

Proof

From (4.4.65) and the standard regularization properties of the Neumann heat semigroup $(e^{\tau \varDelta })_{\tau >0}$ in Winkler (2010), one can conclude that

$$\begin{aligned}&\quad \Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )}\nonumber \\&\le e^{-t}\Vert \nabla e^{t \varDelta }c_0\Vert _{L^\infty (\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-u\cdot \nabla c)(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&\le c_3(1+t^{-\frac{1}{2}})e^{- t}\Vert \nabla c_0\Vert _{L^3(\varOmega )}+\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-m_{\infty })(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\&\quad +\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds. \end{aligned}$$

(4.4.72)

In the second inequality, we have used $\nabla e^{(t-s)(\varDelta -1)}m_{\infty }=0$.

From Lemmas 1.1(ii), 4.41 and 4.3, it follows that

$$\begin{aligned}&\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}(m-m_{\infty })(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert (m-m_{\infty }) (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_2 (M_5+M_1)\varepsilon \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)} (1+s^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-\alpha _1 s}ds\nonumber \\ \le&c_2 c_{10}(M_5+M_1)\varepsilon (1+t^{\min \{0,\frac{1}{2}-\frac{3}{2p_0}\}})e^{-\min \{\alpha _1,\lambda _1+1\}t}\nonumber \\ \le&c_2 c_{10}(M_5+M_1)\varepsilon (1+t^{-\frac{1}{2}})e^{-\alpha _1t}.\nonumber \end{aligned}$$

(4.4.73)

On the other hand, by Lemmas 1.1(ii), 4.3 and the definition of T, we obtain

$$\begin{aligned}&\int _0^t\Vert \nabla e^{(t-s)(\varDelta -1)}u\cdot \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u\cdot \nabla c(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ \le&c_2\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)}\Vert u(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c(\cdot ,s) \Vert _{L^\infty (\varOmega )}ds\nonumber \\ \le&c_2M_3 M_2\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})e^{-(\lambda _1+1)(t-s)} (1+s^{-\frac{1}{2}+\frac{3}{2q_0}}) (1+s^{-\frac{1}{2}}) e^{-(\alpha _1+\alpha _2) s} \nonumber \\ \le&3c_2M_3M_2\varepsilon ^{2}\int _0^te^{-(\lambda _1+1)(t-s)} e^{-(\alpha _1+\alpha _2) s}(1+(t-s)^{-\frac{1}{2}-\frac{3}{2q_0}})(1+s^{-1+\frac{3}{2q_0}})ds\nonumber \\ \le&3c_2M_3M_2c_{10}\varepsilon ^{2}(1+t^{-\frac{1}{2}})e^{-\alpha _1t}. \nonumber \end{aligned}$$

(4.4.74)

Hence, combining the above inequalities and applying (4.4.51) and (4.4.54), we arrive at the desired conclusion.

Lemma 4.45

Under the assumptions of Proposition 4.4, we have

$$\begin{aligned} \Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t} \quad \hbox {for all}\,\,t\in (0,T),\,\theta \in [q_0,\infty ]. \end{aligned}$$

Proof

From (4.4.64), we have

$$\begin{aligned} \rho (\cdot ,t)=&e^{t(\varDelta -m_\infty )}\rho _0-\int _0^te^{(t-s)(\varDelta -m_\infty )}(\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)-u\cdot \nabla \rho )(\cdot ,s)ds\\&+ \int _0^te^{(t-s)(\varDelta -m_\infty )} \rho (m_\infty -m)(\cdot ,s)ds. \end{aligned}$$

By Lemma 1.1, the result in Sect. 2 of Winkler (2010) and $\alpha _1<\min \{\lambda _1,m_\infty \}$, we obtain

$$\begin{aligned}&\Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )} \\ \le&e^{-m_\infty t}(\Vert e^{t\varDelta }(\rho _0-\overline{\rho }_0)\Vert _{L^\theta (\varOmega )}+ \Vert \overline{\rho }_0 \Vert _{L^\theta (\varOmega )}) \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}(u\cdot \nabla \rho )(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}\rho (m_\infty -m)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&c_1(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-(\lambda _1+m_\infty )t}\Vert \rho _0-\overline{\rho }_0\Vert _{L^{p_0}(\varOmega )} +(\min \{1,|\varOmega |\})^{-\frac{1}{p_0}}e^{-m_\infty t}\varepsilon \\&+c_4C_S\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })})e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds\\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty ) }\nabla \cdot (\rho u)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)(\varDelta -m_\infty )}\rho (m_\infty -m)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds\\ \le&(2c_1+(\min \{1,|\varOmega |\})^{-\frac{1}{p_0}}) (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) \varepsilon e^{-\alpha _1t} \\&+c_4C_S\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \\&+c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{\infty }(\varOmega )}\Vert u\Vert _{L^{q_0}(\varOmega )}ds \\&+c_1\int _0^t(1+(t-s)^{-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-m_\infty (t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert m-m_\infty \Vert _{L^{\infty }(\varOmega )}ds. \end{aligned}$$

According to the definition of T, Lemmas 4.44 and 4.3, this shows that

$$\begin{aligned}&\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert \nabla c\Vert _{L^\infty (\varOmega )}ds \nonumber \\ \le&3 M_1M_2\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}e^{-2\alpha _1 s}(1+s^ {-\frac{1}{2}-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) ds\nonumber \\ \le&3c_{10} M_1M_2 \varepsilon ^2 (1+t^{\min \{0,-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{\lambda _1,2\alpha _1\}t}.\nonumber \end{aligned}$$

Similarly, we can also get

$$\begin{aligned}&\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-(\lambda _1+m_\infty )(t-s)}\Vert \rho \Vert _{L^{\infty }(\varOmega )}\Vert u\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\ \le&3 M_1M_3\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}e^{-2\alpha _1 s}(1+s^ {-\frac{1}{2}-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) ds\nonumber \\ \le&3c_{10}M_3 M_1\varepsilon ^2(1+t^{\min \{0,-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{\lambda _1,2\alpha _1\}t}\nonumber \end{aligned}$$

and

$$\begin{aligned}&\int _0^t(1+(t-s)^{-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-m_\infty (t-s)}\Vert \rho \Vert _{L^{q_0}(\varOmega )}\Vert m-m_\infty \Vert _{L^{\infty }(\varOmega )}ds\nonumber \\ \le&3 M_1(M_5+M_1)\varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-m_\infty (t-s)} e^{-2\alpha _1 s}(1+s^{-\frac{3}{p_0}+\frac{3}{2 q_0}}) ds\nonumber \\ \le&3c_{10}M_1(M_5+M_1) \varepsilon ^2(1+t^{\min \{0,-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{m_\infty ,2\alpha _1\}t},\nonumber \end{aligned}$$

where the fact that $q_0\in (3,\frac{3p_0}{2(3-p_0)})$ warrants $-\frac{3}{p_0}+\frac{ 3}{2 q_0}>-1$ is used. Hence, the combination of the above inequalities yields

$$\Vert \rho (\cdot ,t)\Vert _{L^\theta (\varOmega )}\le \frac{M_1}{2}\varepsilon (1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t},$$

thanks to (4.4.60), (4.4.59) and (4.4.56).

Lemma 4.46

Under the assumptions of Proposition 4.4, we have

$$\begin{aligned} \Vert (m-\rho )(\cdot ,t)-e^{t\varDelta }(m_0-\rho _0)\Vert _{L^\theta (\varOmega )}\le \frac{\varepsilon }{2}(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })}) e^{-\alpha _1 t} \end{aligned}$$

for $\theta \in [q_0,\infty ]$, $t\in (0,T)$.

Proof

From (4.4.63) and Lemma 1.1(iv), it follows that

$$\begin{aligned}&\Vert (m-\rho )(\cdot ,t)-e^{t\varDelta }(m_0-\rho _0)\Vert _{L^\theta (\varOmega )} \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }(\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)-u\cdot \nabla (m-\rho ))(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot (\rho \mathscr {S}(\cdot ,\rho ,c)\nabla c)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\&+\int _0^t\Vert e^{(t-s)\varDelta }\nabla \cdot ((m-\rho -m_\infty )u)(\cdot ,s)\Vert _{L^\theta (\varOmega )}ds \\ \le&c_4C_S \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert \rho (\cdot ,s)\Vert _{L^{q_0}(\varOmega )}\Vert \nabla c(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\&+c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert u(m-\rho -m_\infty )(\cdot ,s)\Vert _{L^{q_0}(\varOmega )}ds \\ =:&I_1+I_2. \end{aligned}$$

From the definition of T and (4.4.58), we have

$$\begin{aligned} I_1\le&3c_4C_S M_1 M_2 \varepsilon ^2 \int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)} (1+s^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{q_0})}) e^{-2\alpha _1 s}ds \nonumber \\ \le&3c_4C_Sc_{10} M_1M_2 \varepsilon ^2 (1+t^{\min \{0,-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })\}}) e^{-\min \{\lambda _1,2\alpha _1\}t}\nonumber \\ \le&\frac{\varepsilon }{4}(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1t}.\nonumber \end{aligned}$$

From Lemmas 4.40, 4.42 and (4.4.61), it follows that

$$\begin{aligned} I_2&=c_4\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}\Vert m-\rho -m_\infty \Vert _{L^\infty (\varOmega )}\Vert u\Vert _{L^{q_0}(\varOmega )}ds\nonumber \\&\le c_4M_3 M_5\varepsilon ^{2}\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) e^{-\lambda _1(t-s)}(1+s^{-\frac{3}{2p_0}})e^{-\alpha _1s}\nonumber \\&\quad \cdot (1+s^{-\frac{1}{2}+\frac{3}{2q_0}})e^{-\alpha _2 s}ds\nonumber \\&\le 3c_4M_3 M_5\varepsilon ^{2}\int _0^t(1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0}-\frac{1}{\theta })}) (1+s^{-\frac{1}{2}+\frac{3}{2}(\frac{1}{q_0}-\frac{1}{p_0})})\nonumber \\&\quad \cdot e^{-\lambda _1(t-s)}e^{-(\alpha _1+\alpha _2) s}ds\nonumber \\&\le 3c_{10}c_4M_3 M_5\varepsilon ^{2} e^{-\min \{\lambda _1, \alpha _1+\alpha _2\}t}(1+t^{\min \{0,\frac{3}{2} (\frac{1}{\theta }-\frac{1}{p_0})\}})\nonumber \\&\le \frac{\varepsilon }{4}(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta })})e^{-\alpha _1 t}.\nonumber \end{aligned}$$

Combining the above inequalities, we arrive at

$$ \Vert (\rho -m)(\cdot ,t)-e^{t\varDelta }(\rho _0-m_0)\Vert _{L^\theta (\varOmega )}\le \frac{\varepsilon }{2}(1+t^{-\frac{3}{2}(\frac{1}{p_0}-\frac{1}{\theta } )}) e^{-\alpha _1 t} $$

and thus complete the proof of this lemma.

By the above lemmas, one can see that $T=T_{max}$. We will need two more estimates to show that $T_{max}=\infty$.

Lemma 4.47

Under the assumptions of Proposition 4.4, for all $\beta \in (\frac{3}{4},\min \{\frac{5}{4}-\frac{3}{2q_0},1\})$ there exists $M_6>0$, such that

$$\begin{aligned} \Vert A^\beta u(\cdot ,t)\Vert _{L^2(\varOmega )}\le \varepsilon M_6e^{-\alpha _2 t} \quad \hbox {for}\,\, t\in (t_0,T_{max})\,\hbox { with}\, \,t_0=\min \{\frac{T_{max}}{6},1\}. \end{aligned}$$

Proof

The proof is similar to that of (4.4.40), and thus is omitted here.

Lemma 4.48

Under the assumptions of Proposition 4.4, there exists $M_7>0$, such that $\Vert c(\cdot ,t)-m_\infty \Vert _{L^{\infty }(\varOmega )}\le M_7 e^{-\alpha _2 t}$ for all $(t_0,T_{max})$ with $t_0=\min \{\frac{T_{max}}{6},1\}$.

Proof

We refer the readers to the proof of Lemma 3.24 in Li et al. (2019b).

Proof of Proposition 4.4.  We first show that the solution is global, i.e., $T_{max}=\infty$. To this end, according to the extensibility criterion in Lemma 4.30, it suffices to show that there exists $C>0$, such that for all $t_0<t<T_{max}$

$$ \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert m(\cdot ,t)\Vert _{L^\infty (\varOmega )} +\Vert c(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}+ \Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )}<C.$$

From Lemmas 4.41, 4.45 and 4.47, there exist $K_i>0$, $i=1,2,3,4$, such that

$$\begin{aligned}&\Vert m(\cdot ,t)-m_\infty \Vert _{L^{\infty }(\varOmega )}\le K_1 e^{-\alpha _1 t},~~ \Vert \rho (\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\&\Vert \nabla c(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le K_3 e^{-\alpha _1 t}, ~~\Vert A^{\beta }u(\cdot ,t)\Vert _{L^2(\varOmega )} \le K_4 e^{-\alpha _2 t} \end{aligned}$$

for $t\in (t_0,T_{max})$. Furthermore, Lemma 4.48 implies that $\Vert c(\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3'e^{-\alpha _2t}$ with some $K_3'>0$ for all $t\in (t_0,T_{max})$. Since $D(A^\beta )\hookrightarrow L^\infty (\varOmega )$ with $\beta \in (\frac{3}{4},1)$, it follows from Lemma 4.47 that $\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le K_4 e^{-\alpha _2 t}$ for some $K_4>0$ for all $t\in (t_0,T_{max})$. This completes the proof of Proposition 4.4.

4.4.3 Global Boundedness and Decay for General $\mathscr {S}$

Proof of Theorems for general $\mathscr {S}$. We complete the proofs of our theorems by an approximation procedure (see Cao and Lankeit (2016) for example). In order to make the previous results applicable, we introduce a family of smooth functions $\rho _\eta \in C_0^\infty (\varOmega )$ with $0\le \rho _\eta (x)\le 1$ for $\eta \in (0,1)$, and $\lim _{\eta \rightarrow 0}\rho _\eta (x)=1$, and let $\mathscr {S}_\eta (x,\rho ,c)=\rho _\eta (x)\mathscr {S}(x,\rho ,c).$

Using this definition, we regularize (4.1.13) as follows:

$$\begin{aligned} \left\{ \begin{aligned}&(\rho _\eta )_t+u_\eta \cdot \nabla \rho _\eta =\varDelta \rho _\eta -\nabla \cdot (\rho _\eta \mathscr {S}_\eta (x,\rho _\eta ,c_\eta )\nabla c_\eta )-\rho _\eta m_\eta , \\&(m_\eta )_t+u_\eta \cdot \nabla m_\eta =\varDelta m_\eta -\rho _\eta m_\eta , \\&(c_\eta )_t+u_\eta \cdot \nabla c_\eta =\varDelta c_\eta -c_\eta +m_\eta , \\&(u_\eta )_t+(u_\eta \cdot \nabla ) u_\eta =\varDelta u_\eta -\nabla P_\eta +(\rho _\eta +m_\eta )\nabla \phi ,\quad \nabla \cdot u_\eta =0,\\&\displaystyle \frac{\partial \rho _\eta }{\partial \nu }=\frac{\partial m_\eta }{\partial \nu } =\frac{\partial c_\eta }{\partial \nu }=0,~ u_\eta =0 \end{aligned} \right. \end{aligned}$$

(4.4.75)

with the initial data

$$\begin{aligned} \rho _\eta (x,0)=\rho _0(x),~m_\eta (x,0)=m_0(x),~c(x,0)=c_0(x),~u_\eta (x,0)=u_0(x),\quad x\in \varOmega . \end{aligned}$$

(4.4.76)

It is observed that $\mathscr {S}_\eta$ satisfies the additional condition $\mathscr {S}=0$ on $\partial \varOmega$. Therefore based on the discussion in Sect. 4.4.2, under the assumptions of Theorem 4.4 and Theorem 4.5, the problem (4.4.75)–(4.4.76) admits a global classical solution $(\rho _\eta ,m_\eta ,c_\eta ,u_\eta , P_\eta )$ that satisfies

$$\begin{aligned} \Vert m_\eta (\cdot ,t)-m_\infty \Vert _{L^\infty (\varOmega )}\le K_1e^{-\alpha _1 t},\quad \Vert \rho _\eta (\cdot ,t)-\rho _\infty \Vert _{L^\infty (\varOmega )}\le K_2e^{-\alpha _1 t}, \\ \Vert c_\eta (\cdot ,t)-m_\infty \Vert _{W^{1,\infty }(\varOmega )}\le K_3e^{-\alpha _2t}, \quad \Vert A^\beta u_\eta (\cdot ,t)\Vert _{L^2(\varOmega )}\le K_4 e^{-\alpha _2 t} \end{aligned}$$

for some constants $K_i$, $i=1,2,3,4$, and all $t\ge 0$. Applying a standard procedure such as in Lemmas 5.2 and 5.6 of Cao and Lankeit (2016), one can obtain a subsequence of $\{\eta _j\}_{j\in \mathbb {N}}$ with $\eta _j\rightarrow 0$ as $j\rightarrow \infty$ such that $\rho _{\eta _j}\rightarrow \rho , ~m_{\eta _j}\rightarrow m, ~c_{\eta _j}\rightarrow c, u_{\eta _j}\rightarrow u \quad \hbox {in}~ C_{loc}^{\nu ,\frac{\nu }{2}}(\overline{\varOmega }\times (0,\infty ))$ as $j\rightarrow \infty$ for some $\nu \in (0,1)$. Moreover, by the arguments as in Lemmas 5.7 and 5.8 of Cao and Lankeit (2016), one can also show that $(\rho ,m,c,u, P)$ is a classical solution of (4.1.13) with the decay properties asserted in Theorems 4.4 and 4.5, respectively. The proof of our main results is thus complete.

4.5 Large Time Behavior of Solutions to a Coral Fertilization Model with Nonlinear Diffusion

4.5.1 Regularized Problems

At first, we present a natural notion of weak solvability to (4.1.16), (4.1.22) and (4.1.23).

Definition 4.1

For a quadruple of functions (n, c, v, u),  we call it a global weak solution of (4.1.16), (4.1.22) and (4.1.23), if it fulfills

$$\begin{aligned} \left\{ \begin{aligned}&n\in L^1_{loc}({\bar{\varOmega }}\times [0,\infty )),\\&c\in L^{\infty }_{loc}({\bar{\varOmega }}\times [0,\infty ))\bigcap L^1_{loc}([0,\infty );W^{1,1}(\varOmega )),\\&v\in L^{\infty }_{loc}({\bar{\varOmega }}\times [0,\infty ))\bigcap L^1_{loc}([0,\infty );W^{1,1}(\varOmega )),\\&u\in L^1_{loc}([0,\infty );W^{1,1}_0(\varOmega ;\mathbb {R}^3)), \end{aligned}\right. \end{aligned}$$

(4.5.1)

with $n\ge 0,c\ge 0,v\ge 0$ in $\varOmega \times (0,\infty ),$ and

$$\begin{aligned} \begin{aligned}&E(n),~n |\nabla c|,~n |u|,~c|u|~\text {and}~v |u|~\text {belong to}~L^{1}_{loc}({\bar{\varOmega }}\times [0,\infty )),\end{aligned}\end{aligned}$$

(4.5.2)

where $E(s):=\int ^s_0 D(\sigma )d\sigma ,$ if $\nabla \cdot u=0$ in the distributional sense, if

$$\begin{aligned} \begin{aligned} -\int ^{\infty }_0\int _{\varOmega }n\varphi _t-\int _{\varOmega }n_0\varphi (\cdot ,0) =&-\int ^{\infty }_0\int _{\varOmega }E(n)\varDelta \varphi +\int ^{\infty }_0\int _{\varOmega }n\nabla c\cdot \nabla \varphi \\&+\int ^{\infty }_0\int _{\varOmega }n u\cdot \nabla \varphi -\int ^{\infty }_0\int _{\varOmega }nv \varphi \end{aligned}\end{aligned}$$

(4.5.3)

for any $\varphi \in C^{\infty }_0({\bar{\varOmega }}\times [0,\infty ))$ satisfying $\frac{\partial \varphi }{\partial \nu }=0,$ if

$$\begin{aligned} \begin{aligned} -\int ^{\infty }_0\int _{\varOmega }c\varphi _t-\int _{\varOmega }c_0\varphi (\cdot ,0)=&-\int ^{\infty }_0\int _{\varOmega }\nabla c\cdot \nabla \varphi -\int ^{\infty }_0\int _{\varOmega }c\varphi +\int ^{\infty }_0\int _{\varOmega }v\varphi \\&+\int ^{\infty }_0\int _{\varOmega }cu\cdot \nabla \varphi \end{aligned}\end{aligned}$$

(4.5.4)

and

$$\begin{aligned} \begin{aligned} -\int ^{\infty }_0\int _{\varOmega }v\varphi _t-\int _{\varOmega }v_0\varphi (\cdot ,0)=&-\int ^{\infty }_0\int _{\varOmega }\nabla v\cdot \nabla \varphi -\int ^{\infty }_0\int _{\varOmega }v n\varphi +\int ^{\infty }_0\int _{\varOmega }v u\cdot \nabla \varphi \end{aligned}\end{aligned}$$

(4.5.5)

for any $\varphi \in C^{\infty }_0({\bar{\varOmega }}\times [0,\infty )),$ as well as if

$$\begin{aligned} \begin{aligned} -\int ^{\infty }_0\int _{\varOmega }u\cdot \varphi _t-\int _{\varOmega }u_0\varphi (\cdot ,0)=&-\int ^{\infty }_0\int _{\varOmega }\nabla u\cdot \nabla \varphi +\int ^{\infty }_0\int _{\varOmega }(n+v)\nabla \varPhi \cdot \varphi \end{aligned}\end{aligned}$$

(4.5.6)

for any $\varphi \in C^{\infty }_0\left( {\bar{\varOmega }}\times [0,\infty );\mathbb {R}^3\right)$ fulfilling $\nabla \cdot \varphi \equiv 0.$

Now, in line with the analysis in closely related settings (Winkler 2015b, 2018c), let us introduce a family of regularized problems of (4.1.16), (4.1.22) and (4.1.23) through a standard approximation procedure. Thereupon, the corresponding approximated problems appear as

$$\begin{aligned} \left\{ \begin{aligned}&n_{\varepsilon t}+u_{\varepsilon }\cdot \nabla n_{\varepsilon }=\nabla \cdot (D_{\varepsilon }(n_{\varepsilon }) \nabla n_{\varepsilon })-\nabla \cdot (n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })\nabla c_{\varepsilon })-F_{\varepsilon }(n_{\varepsilon })v_{\varepsilon },\quad x\in \varOmega ,~t>0, \\&c_{\varepsilon t}+u_{\varepsilon }\cdot \nabla c_{\varepsilon }=\varDelta c_{\varepsilon }-c_{\varepsilon }+v_{\varepsilon },\quad x\in \varOmega ,~t>0, \\&v_{\varepsilon t}+u_{\varepsilon }\cdot \nabla v_{\varepsilon }=\varDelta v_{\varepsilon }-v_{\varepsilon } F_{\varepsilon }(n_{\varepsilon }),\quad x\in \varOmega ,~t>0, \\&u_{\varepsilon t}=\varDelta u_{\varepsilon }+\nabla P_{\varepsilon }+(n_{\varepsilon }+v_{\varepsilon })\nabla \varPhi ,\quad \nabla \cdot u_{\varepsilon }=0,\quad x\in \varOmega ,~t>0, \\&\frac{\partial n_{\varepsilon }}{\partial \nu }=\frac{\partial c_{\varepsilon }}{\partial \nu }=\frac{\partial v_{\varepsilon }}{\partial \nu }=0,~u_{\varepsilon }=0,\quad x\in \partial \varOmega ,~t>0, \\&n_{\varepsilon }(x,0)=n_{0}(x),~~c_{\varepsilon }(x,0)=c_{0}(x), ~~v_{\varepsilon }(x,0)=v_{0}(x),~~u_{\varepsilon }(x,0)=u_{0}(x),\quad x\in \varOmega , \end{aligned}\right. \end{aligned}$$

(4.5.7)

where for each $\varepsilon \in (0,1)$ $(D_{\varepsilon })_{\varepsilon \in (0,1)}\in C^2([0,\infty ))$ fulfills

$$\begin{aligned} \begin{aligned} D_{\varepsilon }(s)\ge \varepsilon ~~\text {and}~~ D(s)\le D_{\varepsilon }(s)\le D(s)+2\varepsilon ,~~s\ge 0, \end{aligned}\end{aligned}$$

(4.5.8)

and where

$$\begin{aligned} F_{\varepsilon }(s):=\int ^s_0\rho _{\varepsilon }(\sigma )d\sigma ,\quad s\ge 0~~\text {for all}~~\varepsilon \in (0,1) \end{aligned}$$

(4.5.9)

with $(\rho _{\varepsilon })_{\varepsilon \in (0,1)}\subset C^{\infty }_0([0,\infty ))$ having the properties that for each $\varepsilon \in (0,1)$

$$\begin{aligned} 0\le \rho _{\varepsilon }\le 1~~\text {in}~~[0,\infty ),~~\rho _{\varepsilon }\equiv 1~~\text {in}~~[0,\frac{1}{\varepsilon }] ~~\text {and}~~\rho _{\varepsilon }\equiv 0~~\text {in}~~[\frac{2}{\varepsilon },\infty ), \end{aligned}$$

(4.5.10)

from which and (4.5.9) one can infer that for each $\varepsilon \in (0,1)$

$$\begin{aligned} F_{\varepsilon }\in C^{\infty }([0,\infty )),~~0\le F_{\varepsilon }(s)\le s~~\text {and}~~0\le F'_{\varepsilon }(s)\le 1,~~s\ge 0 \end{aligned}$$

(4.5.11)

as well as

$$\begin{aligned} F_{\varepsilon }(s)\rightarrow s~~\text {and}~~F'_{\varepsilon }(s)\rightarrow 1~~\text {for any}~~s>0~~\text {as}~~\varepsilon \rightarrow 0. \end{aligned}$$

(4.5.12)

Actually, the local solvability of (4.5.7) can be verified through a suitable adaptation of standard fixed point arguments as proceeding in Winkler (2012, Lemma 2.1), so here we merely present the associated assertions.

Lemma 4.49

Let (4.1.17) be fulfilled. Then for any $(n_0,c_0,v_0,u_0)$ complying with (4.1.25) and each $\varepsilon \in (0,1),$ one can find $T_{\max ,\varepsilon }\in (0,+\infty ]$ and functions

$$\begin{aligned} \left\{ \begin{aligned}&n_{\varepsilon }\in C^0({\bar{\varOmega }}\times [0,T_{\max ,\varepsilon }))\bigcap C^{2,1}({\bar{\varOmega }}\times (0,T_{\max ,\varepsilon })),\\&c_{\varepsilon }\in \bigcap _{r>3}C^0([0,T_{\max ,\varepsilon });W^{1,r}(\varOmega ))\bigcap C^{2,1}({\bar{\varOmega }}\times (0,T_{\max ,\varepsilon })),\\&v_{\varepsilon }\in \bigcap _{r>3}C^0([0,T_{\max ,\varepsilon });W^{1,r}(\varOmega ))\bigcap C^{2,1}({\bar{\varOmega }}\times (0,T_{\max ,\varepsilon })),\\&u_{\varepsilon }\in C^0({\bar{\varOmega }}\times [0,T_{\max ,\varepsilon });\mathbb {R}^3)\bigcap C^{2,1}({\bar{\varOmega }}\times (0,T_{\max ,\varepsilon });\mathbb {R}^3), \end{aligned}\right. \end{aligned}$$

(4.5.13)

such that $n_{\varepsilon }\ge 0,c_{\varepsilon }\ge 0$ and $v_{\varepsilon }\ge 0$ in $\varOmega \times [0,T_{\max ,\varepsilon }),$ that with $P_{\varepsilon }\in C^{1,0}(\varOmega \times (0,T_{\max ,\varepsilon }))$ the quintuple of functions $(n_{\varepsilon },c_{\varepsilon },v_{\varepsilon },u_{\varepsilon },P_{\varepsilon })$ forms a classical solution of (4.5.7) in $\varOmega \times [0,T_{\max ,\varepsilon }),$ and that with some $\alpha \in (\frac{3}{4},1)$ either $T_{\max ,\varepsilon }<\infty$ or

$$\begin{aligned} \begin{aligned}&\limsup _{t\nearrow T_{\max ,\varepsilon }}\left\{ \Vert n_{\varepsilon }(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}+\Vert c_{\varepsilon }(\cdot ,t)\Vert _{W^{1,r}(\varOmega )} +\Vert v_{\varepsilon }(\cdot ,t)\Vert _{W^{1,r}(\varOmega )}+\Vert A^{\alpha }u_{\varepsilon }(\cdot ,t)\Vert _{L^2(\varOmega )}\right\} \\&=\infty \end{aligned} \end{aligned}$$

(4.5.14)

holds.

Thanks to the consumption interaction between n and v,  (4.5.7) implies following basic estimates.

Lemma 4.50

Let $M_0:=\max \{\int _{\varOmega }n_0,\int _{\varOmega }v_0,\Vert c_0\Vert _{L^{\infty }(\varOmega )},\Vert v_0\Vert _{L^{\infty }(\varOmega )}\}.$ Then the solutions constructed in Lemma 4.49 satisfy

$$\begin{aligned} \int _{\varOmega }n_{\varepsilon }(\cdot ,t)\le M_0 \quad \text {and}\quad \int _{\varOmega }v_{\varepsilon }(\cdot ,t)\le M_0\end{aligned}$$

(4.5.15)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1),$ as well as

$$\begin{aligned} \Vert v_{\varepsilon }(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le M_0\quad \text {and}\quad \Vert c_{\varepsilon }(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le M_0\end{aligned}$$

(4.5.16)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ Moreover, there exists some $C>0$ such that

$$\begin{aligned} \int ^{\infty }_0\int _{\varOmega } F(n_{\varepsilon })v_{\varepsilon }\le C\quad \text {for all}~~\varepsilon \in (0,1) \end{aligned}$$

(4.5.17)

and

$$\begin{aligned} \int ^{\infty }_0\int _{\varOmega } |\nabla v_{\varepsilon }|^2\le C\quad \text {for all}~~\varepsilon \in (0,1) \end{aligned}$$

(4.5.18)

as well as

$$\begin{aligned} \int ^{\infty }_0\int _{\varOmega } |\nabla c_{\varepsilon }|^2\le C\quad \text {for all}~~\varepsilon \in (0,1). \end{aligned}$$

(4.5.19)

Proof

The detailed process of the derivation thereof can be found in Espejo and Winkler (2018); Liu (2020).

In the final, we provide some conditional estimates of $(u_{\varepsilon })_{\varepsilon \in (0,1)},$ which reveal the relationships between temporally independent estimates of $(u_{\varepsilon })_{\varepsilon \in (0,1)}$ and uniform $L^p~(p>1)$ norms of $(n_{\varepsilon })_{\varepsilon \in (0,1)}.$

Lemma 4.51

uppose $(n_{\varepsilon },c_{\varepsilon },v_{\varepsilon },u_{\varepsilon })_{\varepsilon \in (0,1)}$ are solutions as established in Lemma 4.49. Let $p\ge 2,l>3$ and $\alpha \in (\frac{3}{4},1).$ Then for any $\iota >0,$ one can find $M_1=M_1(p,l,\iota ,M_0)>0$ and $M_2=M_2(\alpha ,p,\iota ,M_0)>0$ such that

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^l(\varOmega )}\le M_1\cdot \left\{ 1+\sup _{\tau \in (0,t)}\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}\right\} ^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) } \end{aligned}$$

(4.5.20)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1),$ and that

$$\begin{aligned} \Vert A^{\alpha }u_{\varepsilon }(\cdot ,t)\Vert _{L^2(\varOmega )}\le M_2\cdot \left\{ 1+\sup _{\tau \in (0,t)}\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}\right\} ^{\frac{p}{p-1}\cdot \left( \frac{4\alpha -1}{6}+\iota \right) } \end{aligned}$$

(4.5.21)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$

Proof

Recalling (Winkler 2021b, Corollary 2.1), we infer from $u_{\varepsilon }$-equation in (4.5.7) that

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^l(\varOmega )}\le C_1\cdot \left\{ 1+\sup _{\tau \in (0,t)}\Vert n_{\varepsilon }(\cdot ,\tau ) +v_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}\right\} ^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) } \end{aligned}$$

(4.5.22)

with some $C_1=C_1(p,l,\iota ,M_0)>0$ for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1),$ where due to (4.5.16),

$$\begin{aligned} \begin{aligned} \Vert n_{\varepsilon }(\cdot ,\tau )+v_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )} \le&\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}+\Vert v_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}\\ \le&\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}+M_0|\varOmega |^{\frac{1}{p}} \end{aligned}\end{aligned}$$

(4.5.23)

for any $\tau \in (0,t)$ with each $t\in (0,T_{\max ,\varepsilon })$ for all $\varepsilon \in (0,1).$ Thereupon, we can rewrite (4.5.22) as

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^l(\varOmega )}\le C_1\cdot \left\{ 1+M_0|\varOmega |^{\frac{1}{p}} +\sup _{\tau \in (0,t)}\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}\right\} ^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) } \end{aligned}$$

(4.5.24)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ With the choice of $M_1:=C_1\cdot (1+M_0|\varOmega |^{\frac{1}{p}}),$ (4.5.20) is implied by (4.5.24). In a flavor quite similar to the reasoning of (4.5.20), (4.5.21) follows from a combination of Winkler (2021b, Proposition 1.1) with (4.5.23).

4.5.2 Conditional Uniform Bounds for $(\nabla c_{\varepsilon })_{\varepsilon \in (0,1)}$

In fact, for the derivation of uniform $L^p$ bounds of $(n_{\varepsilon })_{\varepsilon \in (0,1)},$ besides the $\varepsilon$-independent conditional estimates of $(u_{\varepsilon })_{\varepsilon \in (0,1)}$ as given by Lemma 4.51, it is also essential to gain similar uniform estimates for signal gradients with respect to the temporally independent $L^p$ norms of $(n_{\varepsilon })_{\varepsilon \in (0,1)}$ in accordance with the recursive frameworks established in Winkler (2021b). For convenience in expressions, we make use of the following abbreviations:

$$\begin{aligned} I_{p,\varepsilon }(t):=1+\sup _{\tau \in (0,t)}\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )},\quad t\in (0,T_{\max ,\varepsilon })~~\text {for all}~~\varepsilon \in (0,1) \end{aligned}$$

(4.5.25)

and

$$\begin{aligned} K_{q,\theta ,\varepsilon }(t):=1+\sup _{\tau \in (0,t)}\left\| B^{\theta }\left( c_{\varepsilon }(\cdot ,\tau )-e^{-sB}c_0\right) \right\| _{L^q(\varOmega )},\quad t\in (0,T_{\max ,\varepsilon }) \end{aligned}$$

(4.5.26)

for all $\varepsilon \in (0,1)$.

Lemma 4.52

Let $\theta \in (\frac{1}{2},1)$ and $q>3.$ Then for any $\iota >0$, one can find some $C=C(\theta ,q,\iota )>0$ satisfying

$$\begin{aligned} \left\| \nabla (c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^{\infty }(\varOmega )}\le C\cdot \left\{ 1+\sup _{\tau \in (0,t)}\left\| B^{\theta }(c_{\varepsilon }(\cdot ,\tau )-e^{-\tau B}c_0)\right\| _{L^q(\varOmega )}\right\} ^{\frac{q+3}{2\theta q}+\iota }\end{aligned}$$

(4.5.27)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$

Proof

Since $\theta \in (\frac{1}{2},1)$ enable s us to choose $q>3$ large enough such that $1-\frac{q+3}{2\theta q}>0,$ this ensures the existence of $\iota >0$ sufficiently small such that $\iota <1-\frac{q+3}{2\theta q},$ which allows for the following choice of $\vartheta ,$ namely

$$\begin{aligned} \vartheta (\iota ):=\frac{q+3}{2q}+\iota \theta <\theta . \end{aligned}$$

(4.5.28)

From the interpolation inequality provided by Friedman (1969, Theorem 2.14.1) for fractional powers of sectorial operators, it follows that

$$\begin{aligned} \begin{aligned}&\left\| B^{\vartheta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^q(\varOmega )} \\ \le&C_1\left\| B^{\theta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| ^{\frac{\vartheta }{\theta }}_{L^q(\varOmega )}\left\| c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0\right\| ^{\frac{\theta -\vartheta }{\theta }}_{L^q(\varOmega )}\\ \le&C_1\left\{ 2M_0|\varOmega |^{\frac{1}{q}}\right\} ^{1-\iota -\frac{q+3}{2\theta q}}\left\| B^{\theta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| ^{\frac{q+3}{2\theta q}+\iota }_{L^q(\varOmega )} \end{aligned}\end{aligned}$$

(4.5.29)

with some $C_1=C_1(\theta ,q,\iota )>0$ and $M_0>0$ as taken in Lemma 4.50 for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ Combining with the embedding $D(B^{\vartheta })\hookrightarrow W^{1,\infty }(\varOmega )$ (Henry 1981), we obtain from (4.5.29) that

$$\begin{aligned}&\left\| \nabla (c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^{\infty }(\varOmega )} \\ \le&C_2\left\| B^{\vartheta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^q(\varOmega )}\\ \le&C_2C_1\left\{ 2M_0|\varOmega |^{\frac{1}{q}}\right\} ^{1-\iota -\frac{q+3}{2\theta q}}\left\| B^{\theta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| ^{\frac{q+3}{2\theta q}+\iota }_{L^q(\varOmega )} \end{aligned}$$

with $C_2=C_2(\theta ,q,\iota )>0$ for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1),$ and whereafter (4.5.27) holds with $C:=C_2C_1\left\{ 2M_0|\varOmega |^{\frac{1}{q}}\right\} ^{1-\iota -\frac{q+3}{2\theta q}}$.

With the aid of Lemma 4.52, the following conditional estimates can be established by means of the $L^p$-$L^q$ estimates for fractional powers of sectorial operators (Horstmann and Winkler 2005, (3)).

Lemma 4.53

Let $\theta \in (\frac{1}{2},1),$ and let $q>3,~p\ge 2.$ Then for each $\iota >0$ there exists $C=C(\theta ,q,p,\iota )>0$ such that

$$\begin{aligned} \left\| B^{\theta }(c_{\varepsilon }(\cdot ,\tau )-e^{-\tau B}c_0)\right\| _{L^q(\varOmega )}\le C\cdot \left\{ 1+\sup _{\tau \in (0,t)}\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}\right\} ^{\frac{p}{p-1}\cdot \left( \frac{2\theta }{3}+\iota \right) } \end{aligned}$$

(4.5.30)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$

Proof

Picking $\iota >0$ sufficiently small such that

$$\begin{aligned} \iota <\min \left\{ 1-\frac{q+3}{2\theta q},2q(1-\theta )\right\} , \end{aligned}$$

(4.5.31)

and then letting

$$\begin{aligned} l:=\frac{3q}{3+2q(1-\theta )-\iota },\end{aligned}$$

(4.5.32)

one can observe from $\iota <2q(1-\theta )$ and $q>3+2q-2q\theta +2q\theta \iota >3+2q-2q\theta$ implied by (4.5.31) and (4.5.32), respectively, that

$$\begin{aligned} q=\frac{3q}{3+2q-2q\theta -2q(1-\theta )}>l=\frac{3q}{3+2q-2q\theta -\iota }>\frac{3q}{3+2q-2q\theta }>3. \end{aligned}$$

(4.5.33)

Next, we apply $B^{\theta }$ to the following variation-of-constants representation

$$c_{\varepsilon }(\cdot ,t)-e^{-tB}c_0=\int ^t_0e^{-B(t-\tau )}\left\{ v_{\varepsilon }(\cdot ,\tau )-u_{\varepsilon }(\cdot ,\tau )\nabla c_{\varepsilon }(\cdot ,\tau )\right\} d\tau $$

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1)$ to have

$$\begin{aligned} \begin{aligned}\left\| B^{\theta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^q(\varOmega )} \le&\int ^t_0\left\| B^{\theta }e^{-B(t-\tau )}v_{\varepsilon }(\cdot ,\tau )\right\| _{L^q(\varOmega )}d\tau \\&+\int ^t_0\left\| B^{\theta }e^{-B(t-\tau )}u_{\varepsilon }(\cdot ,\tau )\nabla c_{\varepsilon }(\cdot ,\tau )\right\| _{L^q(\varOmega )}d\tau \end{aligned}\end{aligned}$$

(4.5.34)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ Recalling the $L^p$-$L^q$ estimates for fractional powers of sectorial operators (Horstmann and Winkler, 2005, (3)) and the following regularity features of the Neumman heat semigroup (Henry 1981; Winkler 2010), namely

$$\begin{aligned} \left\| \nabla e^{-tB}c_0\right\| _{L^{\infty }(\varOmega )}\le C_1\left\| \nabla c_0\right\| _{L^{\infty }(\varOmega )} \end{aligned}$$

(4.5.35)

with some $C_1>0,$ we gain from (4.5.16), (4.5.20), (4.5.25), (4.5.26) and (4.5.27) that

$$\begin{aligned} \begin{aligned} \int ^t_0\left\| B^{\theta }e^{-B(t-\tau )}v_{\varepsilon }(\cdot ,\tau )\right\| _{L^q(\varOmega )}d\tau \le&C_2\int ^t_0\left( 1+(t-\tau )^{-\theta }\right) e^{-(t-\tau )}\Vert v_{\varepsilon }(\cdot ,\tau )\Vert _{L^q(\varOmega )}d\tau \\ \le&C_2M_0|\varOmega |^{\frac{1}{q}}\int ^t_0\left( 1+(t-\tau )^{-\theta }\right) e^{-(t-\tau )}d\tau \le C_3 \end{aligned}\end{aligned}$$

(4.5.36)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1)$ with $C_2>0$ and

$$C_3:=C_2M_0|\varOmega |^{\frac{1}{q}}\int ^{\infty }_0\left( 1+\sigma ^{-\theta }\right) e^{-\sigma }d\sigma <\infty $$

thanks to $\theta \in \left( \frac{1}{2},1\right) ,$ and that

$$\begin{aligned} \begin{aligned}&\int ^t_0\left\| B^{\theta }e^{-B(t-\tau )}u_{\varepsilon }(\cdot ,\tau )\nabla c_{\varepsilon }(\cdot ,\tau )\right\| _{L^q(\varOmega )}d\tau \\ \le&C_4\int ^t_0\left( 1+(t-\tau )^{-\theta -\frac{3}{2}(\frac{1}{l}-\frac{1}{q})}\right) e^{-(t-\tau )}\Vert u_{\varepsilon }(\cdot ,\tau )\nabla c_{\varepsilon }(\cdot ,\tau )\Vert _{L^l(\varOmega )}d\tau \\ \le&C_4\int ^t_0\left( 1+(t-\tau )^{-\theta -\frac{3}{2}(\frac{1}{l}-\frac{1}{q})}\right) e^{-(t-\tau )}\Vert u_{\varepsilon }(\cdot ,\tau )\Vert _{L^l(\varOmega )} \Vert \nabla c_{\varepsilon }(\cdot ,\tau )\Vert _{L^{\infty }(\varOmega )}d\tau \\ \le&C_4\int ^t_0\left( 1+(t-\tau )^{-\theta -\frac{3}{2}(\frac{1}{l}-\frac{1}{q})}\right) e^{-(t-\tau )}\Vert u_{\varepsilon }(\cdot ,\tau )\Vert _{L^l(\varOmega )} \\&\cdot \left\{ \left\| \nabla (c_{\varepsilon }(\cdot ,\tau )-e^{-tB}c_0)\right\| _{L^{\infty }(\varOmega )}+\Vert \nabla e^{-tB}c_0\Vert _{L^{\infty }(\varOmega )}\right\} d\tau \\ \le&C_4M_1\int ^t_0\left( 1+(t-\tau )^{-\theta -\frac{3}{2}(\frac{1}{l}-\frac{1}{q})}\right) e^{-(t-\tau )}d\tau \cdot I^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) }_{p,\varepsilon }(t)\\&\cdot \left\{ C_5K^{\frac{q+3}{2\theta q}+\iota }_{q,\theta ,\varepsilon }(t)+C_1\Vert \nabla c_0\Vert _{L^{\infty }(\varOmega )}\right\} \\ \le&C_6I^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) }_{p,\varepsilon }(t)\cdot K^{\frac{q+3}{2\theta q}+\iota }_{q,\theta ,\varepsilon }(t) \end{aligned}\end{aligned}$$

(4.5.37)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1),$ where $C_4,C_5$ are positive constants and $C_6:=C_4M_1(C_5+C_1M_0)\int ^{\infty }_0\left( 1+{\sigma }^{-\theta -\frac{3}{2}(\frac{1}{l}-\frac{1}{q})}\right) e^{-\sigma }d\sigma <\infty$ due to (4.5.33). Inserting (4.5.36) and (4.5.37) into (4.5.34) entails

$$\begin{aligned} \left\| B^{\theta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^q(\varOmega )} \le C_3+C_6I^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) }_{p,\varepsilon }(t)\cdot K^{\frac{q+3}{2\theta q}+\iota }_{q,\theta ,\varepsilon }(t) \end{aligned}$$

(4.5.38)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ It can be readily seen from (4.5.31) that

$$\iota +\frac{q+3}{2\theta q}<1,$$

which allows for an application of Young’s inequality to attain

$$\left\| B^{\theta }(c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^q(\varOmega )} \le C_3+\frac{1}{2}K_{q,\theta ,\varepsilon }(t)+C_7I^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) \cdot \frac{2\theta q}{2\theta q-q-3-2\theta q\iota }}_{p,\varepsilon }(t)$$

with certain $C_7=C_7(\theta ,q,p,l,\iota )$ for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ We thus combine with (4.5.26) to have

$$K_{q,\theta ,\varepsilon }(t)\le 1+C_3+\frac{1}{2}K_{q,\theta ,\varepsilon }(t)+C_7I^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) \cdot \frac{2\theta q}{2\theta q-q-3-2\theta q\iota }}_{p,\varepsilon }(t),$$

namely

$$\begin{aligned} K_{q,\theta ,\varepsilon }(t)\le 2(1+C_3)+2C_7I^{\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota \right) \cdot \frac{2\theta q}{2\theta q-q-3-2\theta q\iota }}_{p,\varepsilon }(t) \end{aligned}$$

(4.5.39)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ Setting

$$\psi (\tilde{\iota }):=\frac{p}{p-1}\cdot \left( \frac{2\theta q-q-3+\tilde{\iota }}{3q}+\tilde{\iota }\right) \cdot \frac{2\theta q}{2\theta q-q-3-2\theta q\tilde{\iota }},$$

we note that $\psi (\tilde{\iota })\searrow \frac{p}{p-1}\cdot \frac{2\theta }{3}$ as $\tilde{\iota }\searrow 0,$ whereupon for arbitrarily small $\iota >0$ one can pick $\iota '\in \left( 0,\min \left\{ 1-\frac{q+3}{2\theta q},2q(1-\theta )\right\} \right)$ such that

$$\psi (\iota ')\le \frac{p}{p-1}\cdot \frac{2\theta }{3}+\iota .$$

An elementary calculation along with (4.5.32) thus shows

$$\begin{aligned}&\frac{p}{p-1}\cdot \left( \frac{l-3}{3l}+\iota '\right) \cdot \frac{2\theta q}{2\theta q-q-3-2\theta q\iota '} \\ =&\frac{p}{p-1}\cdot \left( \frac{2\theta q-q-3+\iota '}{3q}+\iota '\right) \cdot \frac{2\theta q}{2\theta q-q-3-2\theta q\iota '}\\ =&\psi (\iota ')\le \frac{p}{p-1}\cdot \frac{2\theta }{3}+\iota ,\end{aligned}$$

which in conjunction with (4.5.39), (4.5.25) and (4.5.26) yields (4.5.30).

With Lemmas 4.52–4.53 at hand, we are in the position to derive the desired conditional uniform $L^{\infty }$ estimates for $(\nabla c_{\varepsilon })_{\varepsilon \in (0,1)}$ from a well-known continuous embedding.

Lemma 4.54

Suppose that $p\ge 2.$ Then for any $\iota >0,$ one can find $C(p,\iota )>0$ fulfilling

$$\begin{aligned} \left\| \nabla c_{\varepsilon }(\cdot ,t)\right\| _{L^{\infty }(\varOmega )}\le C\cdot \left\{ 1+\sup _{\tau \in (0,t)}\Vert n_{\varepsilon }(\cdot ,\tau )\Vert _{L^p(\varOmega )}\right\} ^{\frac{p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) } \end{aligned}$$

(4.5.40)

for any $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$

Proof

For given $\iota >0,$ there exists $q>3$ sufficiently large satisfying $\frac{1}{q}<\iota ,$ which shows

$$\frac{q+3}{3q}<\frac{1}{3}+\iota .$$

Define

$$\phi (\tilde{\iota }):=\left( \frac{q+3}{2\theta q}+\tilde{\iota }\right) \cdot \left( \frac{2\theta }{3}+\tilde{\iota }\right) ,\qquad \tilde{\iota }>0.$$

We can readily see that

$$\phi (\tilde{\iota })\searrow \frac{q+3}{2\theta q}\cdot \frac{2\theta }{3}=\frac{q+3}{3q}<\frac{1}{3}+\iota \quad \text {as}\quad \tilde{\iota }\searrow 0.$$

This enables us to pick some $\iota ''=\iota ''(\iota )>0$ such that

$$\begin{aligned} \phi (\iota '')\le \frac{1}{3}+\iota . \end{aligned}$$

(4.5.41)

Now, from Lemmas 4.52–4.53, we are able to find certain $C_1=C_1(p,q,\theta ,\iota '')>0$ fulfilling

$$\begin{aligned} \left\| \nabla (c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^{\infty }(\varOmega )}\le C_1I^{\frac{p}{p-1}\cdot \left( \frac{2\theta }{3}+\iota ''\right) \cdot \left( \frac{q+3}{2\theta q}+\iota ''\right) }_{p,\varepsilon } \end{aligned}$$

(4.5.42)

for any $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ Apart from that, (4.5.35) provides some $C_2>0$ satisfying

$$\begin{aligned} \left\| \nabla e^{-tB}c_0\right\| _{L^{\infty }(\varOmega )}\le C_2\left\| \nabla c_0\right\| _{L^{\infty }(\varOmega )}. \end{aligned}$$

(4.5.43)

Thereupon, it can be deduced from (4.5.25), (4.5.41), (4.5.42) and (4.5.43) that

$$\begin{aligned}\left\| \nabla c_{\varepsilon }(\cdot ,t)\right\| _{L^{\infty }(\varOmega )}\le&\left\| \nabla (c_{\varepsilon }(\cdot ,t)-e^{-\tau B}c_0)\right\| _{L^{\infty }(\varOmega )}+\left\| \nabla e^{-\tau B}c_0\right\| _{L^{\infty }(\varOmega )}\\ \le&C_1I^{\frac{p}{p-1}\cdot \left( \frac{2\theta }{3}+\iota ''\right) \cdot \left( \frac{q+3}{2\theta q}+\iota ''\right) }_{p,\varepsilon }(t)+C_2\left\| \nabla c_0\right\| _{L^{\infty }(\varOmega )}\\ \le&C_3I^{\frac{p}{p-1}\cdot \left( \frac{2\theta }{3}+\iota ''\right) \cdot \left( \frac{q+3}{2\theta q}+\iota ''\right) }_{p,\varepsilon }(t)\\ =&C_3I^{\frac{p}{p-1}\cdot \phi (\iota '')}_{p,\varepsilon }(t)\\ \le&C_3I^{\frac{p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) }_{p,\varepsilon }(t)\end{aligned}$$

with $C_3:=C_1+C_2\left\| \nabla c_0\right\| _{L^{\infty }(\varOmega )}$ for any $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1),$ as claimed.

4.5.3 A Prior Estimates

Relying on the basic estimates and the conditional estimates obtained in previous sections, we can achieve the boundedness of $(n_{\varepsilon })_{\varepsilon \in (0,1)}$ in temporally independent $L^p$-topology under a milder assumption on m as compared to that imposed in Liu (2020).

Lemma 4.55

Let $m>1.$ Then for any $p>1$ there exists $C=C(p)>0$ such that

$$\begin{aligned} \Vert n_{\varepsilon }(\cdot ,t)\Vert _{L^p(\varOmega )}\le C \end{aligned}$$

(4.5.44)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ In particular, for $p=m,$ one can find $C_{*}>0$ fulfilling

$$\begin{aligned} \int ^T_0\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2\le C_{*}(T+1) \end{aligned}$$

(4.5.45)

for any $T\in (0,T_{\max ,\varepsilon }).$

Proof

Thanks to the arbitrariness of $p>1,$ herein without loss of generality, we let

$$\begin{aligned} p>m. \end{aligned}$$

(4.5.46)

In addition, since $m>1,$ it is possible to choose $\iota >0$ sufficiently small such that

$$\begin{aligned} \lambda :=\frac{1+3\iota }{3m-2}<1. \end{aligned}$$

(4.5.47)

In view of (4.1.24), (4.5.11), $\nabla \cdot u_{\varepsilon }=0$ and the nonnegativity of $n_{\varepsilon }$ and $v_{\varepsilon },$ we test $n_{\varepsilon }$-equation in (4.5.7) by $p n^{p-1}_{\varepsilon }$ and invoke Young’s inequality to have

$$\begin{aligned} \begin{aligned}\frac{d}{dt}\int _{\varOmega }n^{p}_{\varepsilon } =&-p(p-1)\int _{\varOmega }D_{\varepsilon }(n_{\varepsilon })n^{p-2}_{\varepsilon }|\nabla n_{\varepsilon }|^2+p(p-1)\int _{\varOmega }n^{p-1}_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })\nabla n_{\varepsilon }\cdot \nabla c_{\varepsilon }\\&-p\int _{\varOmega }n^{p-1}_{\varepsilon }F_{\varepsilon }(n_{\varepsilon })v_{\varepsilon }\\ \le&-C_Dp(p-1)\int _{\varOmega }n^{m+p-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2+p(p-1)\int _{\varOmega }n^{p-1}_{\varepsilon }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|\\ \le&-\frac{C_Dp(p-1)}{2}\int _{\varOmega }n^{m+p-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2+\frac{p(p-1)}{2C_D}\int _{\varOmega }n^{p-m+1}_{\varepsilon }|\nabla c_{\varepsilon }|^2\\ =&-\frac{2C_Dp(p-1)}{(m+p-1)^2}\int _{\varOmega }\left| \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\right| ^2+\frac{p(p-1)}{2C_D}\int _{\varOmega }n^{p-m+1}_{\varepsilon }|\nabla c_{\varepsilon }|^2\end{aligned}\end{aligned}$$

(4.5.48)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ For the rightmost integral, we deduce from (4.5.25), (4.5.46) and Lemma 4.54 that

$$\begin{aligned} \begin{aligned}&\frac{p(p-1)}{2C_D}\int _{\varOmega }n^{p-m+1}_{\varepsilon }|\nabla c_{\varepsilon }|^2 \\ =&\frac{p(p-1)}{2C_D}\int _{\{n_{\varepsilon }\le 1\}}n^{p-m+1}_{\varepsilon }|\nabla c_{\varepsilon }|^2+\frac{p(p-1)}{2C_D}\int _{\{n_{\varepsilon }>1\}}n^{p-m+1}_{\varepsilon }|\nabla c_{\varepsilon }|^2\\ \le&\frac{p(p-1)|\varOmega |}{2C_D}\Vert \nabla c\Vert ^2_{L^{\infty }(\varOmega )}+\frac{p(p-1)}{2C_D}\Vert \nabla c\Vert ^2_{L^{\infty }(\varOmega )}\int _{\varOmega }n^{p-m+1}_{\varepsilon }\\ \le&\frac{p(p-1)|\varOmega |}{2C_D}I^{\frac{2p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) }_{p,\varepsilon }(t) +\frac{p(p-1)}{2C_D}I^{\frac{2p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) }_{p,\varepsilon }(t)\cdot \int _{\varOmega }n^{p-m+1}_{\varepsilon } \end{aligned}\end{aligned}$$

(4.5.49)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ Since (4.5.46) implies

$$\frac{2}{m+p-1}<\frac{2(p-m+1)}{m+p-1}<6,$$

with $a:=\frac{3(p-m)(m+p-1)}{(p-m+1)(3m+3p-4)}\in (0,1)$ an application of the Gagliardo–Nirenberg inequality combined with (4.5.15) shows that

$$\begin{aligned} \begin{aligned}&\int _{\varOmega }n^{p-m+1}_{\varepsilon }\\ =&\Big \Vert n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert ^{\frac{2(p-m+1)}{m+p-1}}_{L^{\frac{2(p-m+1)}{m+p-1}}(\varOmega )}\\ \le&C_1\left\{ \Big \Vert \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert ^a_{L^2(\varOmega )} \Big \Vert n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert ^{1-a}_{L^{\frac{2}{m+p-1}}(\varOmega )} +\Big \Vert n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert _{L^{\frac{2}{m+p-1}}(\varOmega )}\right\} ^{\frac{2(p-m+1)}{m+p-1}}\\ \le&C_2\Big \Vert \nabla n^{\frac{p+m-1}{2}}_{\varepsilon }\Big \Vert ^{2\cdot \frac{3(p-m)}{3m+3p-4}}_{L^2(\varOmega )}+C_2 \end{aligned}\end{aligned}$$

(4.5.50)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1),$ where both $C_1$ and $C_2$ are positive constants. Observing that

$$\frac{3(p-m)}{3m+3p-4}-1=\frac{-6m+4}{3m+3p-4}<0$$

due to $m>1,$ we again employ Young’s inequality and derive from (4.5.49) and (4.5.50) that

$$\begin{aligned} \begin{aligned}&\frac{p(p-1)}{2C_D}\int _{\varOmega }n^{p-m+1}_{\varepsilon }|\nabla c_{\varepsilon }|^2\\ \le&\frac{p(p-1)|\varOmega |}{2C_D}I^{\frac{2p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) }_{p,\varepsilon }(t) +\frac{p(p-1)C_2}{2C_D}I^{\frac{2p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) }_{p,\varepsilon }(t)\\&+\frac{p(p-1)C_2}{2C_D}I^{\frac{2p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) }_{p,\varepsilon }(t)\cdot \Big \Vert \nabla n^{\frac{p+m-1}{2}}_{\varepsilon }\Big \Vert ^{2\cdot \frac{3(p-m)}{3m+3p-4}}_{L^2(\varOmega )}\\ \le&\frac{p(p-1)(|\varOmega |+C_2)}{2C_D}I^{\frac{2p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) }_{p,\varepsilon }(t) +\frac{C_Dp(p-1)}{(m+p-1)^2}\int _{\varOmega }\left| \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\right| ^2\\&+C_3I^{\frac{p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) \cdot \frac{3m+3p-4}{3m-2}}_{p,\varepsilon }(t) \end{aligned}\end{aligned}$$

(4.5.51)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ In light of the facts that $I_{p,\varepsilon }\ge 1$ and that $I_{p,\varepsilon }$ is nondecreasing with respect to t,  it follows from (4.5.48) and (4.5.51) that

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }n^{p}_{\varepsilon }+\frac{C_Dp(p-1)}{(m+p-1)^2}\int _{\varOmega }\left| \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\right| ^2\le C_4I^{\frac{p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) \cdot \frac{3m+3p-4}{3m-2}}_{p,\varepsilon }(t) \end{aligned}$$

(4.5.52)

with $C_4:=\frac{p(p-1)(|\varOmega |+C_2)}{2C_D}+C_3$ for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ From $m>1$ and (4.5.46), it is clear that $p>1$ and $p>\frac{3}{2}(1-m),$ which warrants that

$$\frac{2}{m+p-1}<\frac{2p}{m+p-1}<6,$$

whence letting $b:=\frac{3(p-1)(m+p-1)}{p(3m+3p-4)}\in (0,1),$ we once more make use of the Gagliardo–Nirenberg inequality to have

$$\begin{aligned}&\left\{ \int _{\varOmega }n^p_{\varepsilon }\right\} ^{\frac{3m+3p-4}{3(p-1)}}\\ =&\Big \Vert n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert ^{\frac{2p}{m+p-1}\cdot \frac{3m+3p-4}{3(p-1)}}_{L^{\frac{2p}{m+p-1}}(\varOmega )}\\ \le&C_5\left\{ \Big \Vert \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert ^b_{L^2(\varOmega )} \Big \Vert n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert ^{1-b}_{L^{\frac{2}{m+p-1}}(\varOmega )} +\Big \Vert n^{\frac{m+p-1}{2}}_{\varepsilon }\Big \Vert _{L^{\frac{2}{m+p-1}}(\varOmega )}\right\} ^{\frac{2p}{m+p-1}\cdot \frac{3m+3p-4}{3(p-1)}}\\ \le&C_6\Big \Vert \nabla n^{\frac{p+m-1}{2}}_{\varepsilon }\Big \Vert ^2_{L^2(\varOmega )}+C_6 \end{aligned}$$

with $C_5>0$ and $C_6>0$ for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1),$ that is

$$\begin{aligned} \int _{\varOmega }\left| \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\right| ^2 \ge \frac{1}{C_6}\left\{ \int _{\varOmega }n^p_{\varepsilon }\right\} ^{\frac{3m+3p-4}{3(p-1)}}-1 \end{aligned}$$

(4.5.53)

for each $t\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ Also due to $I_{p,\varepsilon }\ge 1$ and its nondecreasing features, for each fixed $T\in (0,T_{\max ,\varepsilon }),$ a combination of (4.5.52) with (4.5.53) entails

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }n^{p}_{\varepsilon }+C_7\left\{ \int _{\varOmega }n^p_{\varepsilon }\right\} ^{\frac{3m+3p-4}{3(p-1)}}\le C_8I^{\frac{p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) \cdot \frac{3m+3p-4}{3m-2}}_{p,\varepsilon }(T) \end{aligned}$$

(4.5.54)

with $C_7:=\frac{C_D p(p-1)}{C_6(m+p-1)^2}$ and $C_8:=C_4+\frac{C_D p(p-1)}{(m+p-1)^2}$ for each $t\in (0,T)$ and all $\varepsilon \in (0,1).$ By means of an ODE comparison argument, we obtain from (4.5.54) that for any fixed $T\in (0,T_{\max ,\varepsilon })$

$$\int _{\varOmega }n^p_{\varepsilon }\le \max \left\{ \int _{\varOmega }n^p_0,\left\{ \frac{C_8}{C_7} I^{\frac{p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) \cdot \frac{3m+3p-4}{3m-2}}_{p,\varepsilon }(T)\right\} ^{\frac{3(p-1)}{3m+3p-4}}\right\} $$

for each $t\in (0,T)$ and all $\varepsilon \in (0,1),$ which further implies

$$\begin{aligned} \int _{\varOmega }n^p_{\varepsilon }\le C_9\cdot I^{p\cdot \frac{1+3\iota }{3m-2}}_{p,\varepsilon }(T) =C_9\cdot I^{p\lambda }_{p,\varepsilon }(T) \end{aligned}$$

(4.5.55)

for each $t\in (0,T)$ and all $\varepsilon \in (0,1),$ where $C_9:=\max \left\{ \int _{\varOmega }n^p_0,\left\{ \frac{C_8}{C_7}\right\} ^{\frac{3(p-1)}{3m+3p-4}}\right\} .$ Recalling (4.5.25), one can infer from (4.5.55) that

$$I_{p,\varepsilon }(T)\le 1+C^{\frac{1}{p}}_9I^{\lambda }_{p,\varepsilon }(T) \le C_{10}I^{\lambda }_{p,\varepsilon }(T)$$

with $C_{10}:=1+C^{\frac{1}{p}}_9$ for each $T\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1).$ In view of (4.5.47), this further shows

$$\begin{aligned} I_{p,\varepsilon }(T)\le C^{\frac{1}{1-\lambda }}_{10}\end{aligned}$$

(4.5.56)

for each $T\in (0,T_{\max ,\varepsilon })$ and all $\varepsilon \in (0,1),$ and thus (4.5.44) holds. Combining (4.5.56) with (4.5.52) and (4.5.47) entails

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }n^{p}_{\varepsilon }+\frac{C_Dp(p-1)}{(m+p-1)^2}\int _{\varOmega }\left| \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\right| ^2\le C_{11} \end{aligned}$$

(4.5.57)

with $C_{11}:=C_4\cdot C^{\frac{p}{p-1}\cdot \left( \frac{1}{3}+\iota \right) \cdot \frac{3m+3p-4}{3m-3-3\iota }}_{10}$ for any $t\in (0,T_{\max ,\varepsilon }),$ whence upon an integration of (4.5.57) on (0, T) for each $T\in (0,T_{\max ,\varepsilon })$, we have

$$\begin{aligned} \int _{\varOmega }n^{p}_{\varepsilon }(\cdot ,T)+\frac{C_Dp(p-1)}{(m+p-1)^2}\int ^T_0\int _{\varOmega }\left| \nabla n^{\frac{m+p-1}{2}}_{\varepsilon }\right| ^2\le C_{11}T+\int _{\varOmega }n^p_0. \end{aligned}$$

(4.5.58)

Thanks to the nonnegativity of $n_{\varepsilon },m>1$ and (4.1.25), we let $p=m$ and derive from (4.5.58) that

$$\begin{aligned} \int ^T_0\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2\le C_{12}(T+1) \end{aligned}$$

(4.5.59)

with $C_{12}:=\frac{4}{C_Dm(m-1)}\max \{C_{11},\int _{\varOmega }n^m_0\}$ for each $T\in (0,T_{\max ,\varepsilon }),$ which shows (4.5.45) by choosing $C_{*}=C_{12}$ and thus completes the proof.

Now, we are able to verify the uniform boundedness for the left-hand side of (4.5.14) so as to establish the global solvability of the approximated problems (4.5.7), which underlies the derivation of global boundedness and stabilization in problem (4.1.16), (4.1.22) and (4.1.23) by means of well-established arguments.

Lemma 4.56

Let $m>1.$ Then the family of the solutions $(n_{\varepsilon },c_{\varepsilon },v_{\varepsilon },u_{\varepsilon })_{\varepsilon \in (0,1)}$ as established in Lemma 4.49 solves (4.5.7) globally and has the properties that for any $r>3$ and all $t>0$ there exists $C=C(r)>0$ independent of $\varepsilon \in (0,1)$ such that

$$\begin{aligned} \Vert n_{\varepsilon }(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}+\Vert c_{\varepsilon }(\cdot ,t)\Vert _{W^{1,r}(\varOmega )} +\Vert v_{\varepsilon }(\cdot ,t)\Vert _{W^{1,r}(\varOmega )}+\Vert A^{\alpha }u_{\varepsilon }(\cdot ,t)\Vert _{L^2(\varOmega )}\le C. \end{aligned}$$

(4.5.60)

Proof

At first, for any $l>3$ and each $\alpha \in (\frac{3}{4},1),$ a combination of Lemma 4.55 with Lemma 4.51 provides some $C_1>0$ such that

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^l(\varOmega )}+\Vert A^{\alpha }u_{\varepsilon }(\cdot ,t)\Vert _{L^2(\varOmega )}\le C_1 \end{aligned}$$

(4.5.61)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ Moreover, from (4.5.16), Lemmas 4.54 and 4.55, we can infer the existence of $C_2>0$ fulfilling

$$\begin{aligned} \Vert c_{\varepsilon }(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le C_2 \end{aligned}$$

(4.5.62)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ In conjunction with (4.5.61) and (4.5.62), an application of a Moser-type iteration reasoning (Tao and Winkler 2012a, Lemma A.1) to $n_{\varepsilon }$-equation in (4.5.7) yields

$$\begin{aligned} \Vert n_{\varepsilon }(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le C_3\end{aligned}$$

(4.5.63)

with some $C_3>0$ for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ Apart from that, for any $r>3$ (Liu 2020, Lemma 5.1) combined with (4.5.16) allows for a choice of $C_4>0$ such that

$$\begin{aligned} \Vert v_{\varepsilon }(\cdot ,t)\Vert _{W^{1,r}(\varOmega )} \le C_4 \end{aligned}$$

(4.5.64)

for all $t\in (0,T_{\max ,\varepsilon })$ and $\varepsilon \in (0,1).$ As a result, a collection of (4.5.61)–(4.5.64) along with (4.5.14) shows the global solvability of (4.5.7) and the validity of (4.5.60).

4.5.4 Global Solvability

The task of this section is to construct global weak solutions of (4.1.16), (4.1.22) and (4.1.23) in the sense of Definition 4.1. As the first step toward this, some further regularity features of $(n_{\varepsilon },c_{\varepsilon },v_{\varepsilon },u_{\varepsilon })_{\varepsilon \in (0,1)}$ are essential to be provided.

Lemma 4.57

There exists $\nu \in (0,1)$ with the properties that one can find some $\varepsilon$-independent $C>0$ fulfilling

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{C^{\nu }({\bar{\varOmega }})}\le C\qquad \text {for all}\quad t\ge 0, \end{aligned}$$

(4.5.65)

$$\begin{aligned} \Vert c_{\varepsilon }\Vert _{C^{\nu }({\bar{\varOmega }}\times [t,t+1])}\le C\qquad \text {for all}\quad t\ge 0 \end{aligned}$$

(4.5.66)

and

$$\begin{aligned} \Vert v_{\varepsilon }\Vert _{C^{\nu }({\bar{\varOmega }}\times [t,t+1])}\le C\qquad \text {for all}\quad t\ge 0, \end{aligned}$$

(4.5.67)

and that for any $\tau >0,$ there exists $\varepsilon$-independent $C(\tau )>0$, such that

$$\begin{aligned} \Vert \nabla c_{\varepsilon }\Vert _{C^{\nu }({\bar{\varOmega }}\times [t,t+1])}\le C(\tau )\qquad \text {for all}\quad t\ge \tau \end{aligned}$$

(4.5.68)

and

$$\begin{aligned} \Vert \nabla v_{\varepsilon }\Vert _{C^{\nu }({\bar{\varOmega }}\times [t,t+1])}\le C(\tau )\qquad \text {for all}\quad t\ge \tau . \end{aligned}$$

(4.5.69)

Proof

According to the arguments of Liu (2020, Lemmas 5.4–5.6), (4.5.66)–(4.5.69) can be derived from a combination of maximal Sobolev regularity with appropriate embedding consequences, while (4.5.65) is an immediate result of (4.5.60) because of the embedding $D(A^{\alpha })\hookrightarrow C^{\nu }({\bar{\varOmega }})$ for each $\nu \in \left( 0,2\alpha -\frac{3}{2}\right)$ (Giga 1981; Henry 1981), due to $\alpha \in \left( \frac{3}{4},1\right)$ required by (4.1.25).

In order to take limit of $(n_{\varepsilon })_{\varepsilon \in (0,1)}$ by suitable extraction procedures in the sequel, it is also necessary to explore the regularity properties of time derivatives of $(n_{\varepsilon })_{\varepsilon \in (0,1)}.$ For expressing conveniently, throughout the sequel, we let

$$\begin{aligned} B_n:=\sup _{\varepsilon \in (0,1)}\Vert n_{\varepsilon }\Vert _{L^{\infty }(\varOmega \times (0,\infty ))}. \end{aligned}$$

(4.5.70)

Lemma 4.58

Let $m>1.$ Then for each $T>0,$ there exists $C=C(T)>0$ satisfying

$$\begin{aligned} \int ^T_0\Vert \partial _tn^m_{\varepsilon }(\cdot ,t)\Vert _{(W^{1,\infty }_0(\varOmega ))^{*}}dt\le C(T)\qquad \text {for all}\quad \varepsilon \in (0,1).\end{aligned}$$

(4.5.71)

Furthermore, one can find $C>0$ independent of $\varepsilon \in (0,1)$, such that

$$\begin{aligned} \Vert n_{\varepsilon }(\cdot ,t)-n_{\varepsilon }(\cdot ,s)\Vert _{(W^{2,2}_0(\varOmega ))^{*}}\le C|t-s|\quad \text {for all}~~t\ge 0~~\text {and}~~s\ge 0.\end{aligned}$$

(4.5.72)

Proof

For any fixed $\psi \in C^{\infty }_0({\bar{\varOmega }})$ and $t\in (0,T),$ integrations by parts combined with applications of Young’s inequality on the basis of the first equation in (4.5.7) entails

$$\begin{aligned}&\left| \frac{1}{m}\int _{\varOmega }\partial _tn^m_{\varepsilon }(\cdot ,t)\cdot \psi \right| \\ =&\left| \int _{\varOmega }n^{m-1}_{\varepsilon }\left\{ \nabla \cdot \left( D_{\varepsilon }(n_{\varepsilon })\nabla n_{\varepsilon }-n_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon }) \nabla c_{\varepsilon }-n_{\varepsilon }u_{\varepsilon }\right) -F_{\varepsilon }(n_{\varepsilon })v_{\varepsilon }\right\} \cdot \psi \right| \\ =&\left| -(m-1)\int _{\varOmega }n^{m-2}_{\varepsilon }D_{\varepsilon }(n_{\varepsilon })|\nabla n_{\varepsilon }|^2\psi -\int _{\varOmega }n^{m-1}_{\varepsilon }D_{\varepsilon }(n_{\varepsilon })\nabla n_{\varepsilon }\cdot \nabla \psi \right. \\&+(m-1)\int _{\varOmega }n^{m-1}_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon })\left( \nabla n_{\varepsilon }\cdot \nabla c_{\varepsilon }\right) \psi +\int _{\varOmega }n^{m}_{\varepsilon }F'_{\varepsilon }(n_{\varepsilon }) \nabla c_{\varepsilon }\cdot \nabla \psi \\&\left. +\frac{1}{m}\int _{\varOmega }n^{m}_{\varepsilon }u_{\varepsilon }\cdot \nabla \psi -\int _{\varOmega }n^{m-1}_{\varepsilon }F_{\varepsilon }(n_{\varepsilon })v_{\varepsilon }\psi \right| \\ \le&\left\{ C_D(m-1)\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2+C_D\int _{\varOmega }n^{2m-2}_{\varepsilon }|\nabla n_{\varepsilon }|\right. \\&+(m-1)\int _{\varOmega }n^{m-1}_{\varepsilon }|\nabla n_{\varepsilon }|\cdot |\nabla c_{\varepsilon }|+\int _{\varOmega }n^{m}_{\varepsilon } |\nabla c_{\varepsilon }|\\&\left. +\frac{1}{m}\int _{\varOmega }n^{m}_{\varepsilon }|u_{\varepsilon }| +\int _{\varOmega }n^{m}_{\varepsilon }v_{\varepsilon }\right\} \cdot \Vert \psi \Vert _{W^{1,\infty }(\varOmega )}\\ \le&\left\{ C_D(m-1)\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2+C_D\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2+C_D\int _{\varOmega }n^2_{\varepsilon }\right. \\&+(m-1)\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2+(m-1)\int _{\varOmega }n_{\varepsilon }|\nabla c_{\varepsilon }|^2+\int _{\varOmega }n^{m}_{\varepsilon } |\nabla c_{\varepsilon }|\\&\left. +\frac{1}{m}\int _{\varOmega }n^{m}_{\varepsilon }|u_{\varepsilon }| +\int _{\varOmega }n^{m}_{\varepsilon }v_{\varepsilon }\right\} \cdot \Vert \psi \Vert _{W^{1,\infty }(\varOmega )}\\ \le&\left\{ (C_Dm+m-1)\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2+C_DB^2_n|\varOmega |+(m-1)B_n\int _{\varOmega } |\nabla c_{\varepsilon }|^2\right. \\&\left. +B^{m}_n\int _{\varOmega } |\nabla c_{\varepsilon }| +\frac{B^{m}_n}{m}\int _{\varOmega }|u_{\varepsilon }| +B^{m}_n M_0|\varOmega |\right\} \cdot \Vert \psi \Vert _{W^{1,\infty }(\varOmega )}\end{aligned}$$

for all $\varepsilon \in (0,1),$ with $C_D$ and $M_0$ given by (4.1.24) and (4.5.16), respectively, which thus together with (4.5.45) and (4.5.65) yields (4.5.71). As for (4.5.72), readers can refer to Liu (2020) for its proof.

Now, we are in the position to verify global solvability of (4.1.16), (4.1.22) and (4.1.23).

Lemma 4.59

Let $m>1.$ Then one can find $(\varepsilon _j)_{j\in \mathbb {N}}\subset (0,1),$ a null set $\aleph \subset (0,\infty )$ and functions n, c, v and u complying with (4.5.1) and (4.5.2), such that $\varepsilon _j\searrow 0$ as $j\rightarrow \infty ,$ that $n\ge 0,c\ge 0$ and $v\ge 0$ in $\varOmega \times (0,\infty ),$ and that as $\varepsilon =\varepsilon _j\searrow 0,$ we have

$$\begin{aligned} n_{\varepsilon }\rightarrow n\qquad \text {a.e. in}~\varOmega ~\text {for each}~t\in (0,\infty )\backslash \aleph ,\end{aligned}$$

(4.5.73)

$$\begin{aligned} n_{\varepsilon }{\mathop {\rightharpoonup }\limits ^{*}} n\qquad \text {in}~L^{\infty }(\varOmega \times (0,\infty )),\end{aligned}$$

(4.5.74)

$$\begin{aligned} n_{\varepsilon }\rightarrow n\qquad \text {in}~C^{0}_{loc}\Big ([0,\infty );(W^{2,2}_0(\varOmega ))^{*}\Big ),\end{aligned}$$

(4.5.75)

$$\begin{aligned} c_{\varepsilon }\rightarrow c~~\text {in}~~C^0_{loc}\left( {\bar{\varOmega }}\times [0,\infty )\right) ,\end{aligned}$$

(4.5.76)

$$\begin{aligned} c_{\varepsilon }{\mathop {\rightharpoonup }\limits ^{*}} c~~\text {in}~~L^{\infty }((0,\infty );W^{1,r}(\varOmega ))~~\text {for each}~~r\in (1,\infty ),\end{aligned}$$

(4.5.77)

$$\begin{aligned} \nabla c_{\varepsilon }\rightarrow \nabla c~~\text {in}~~C^0_{loc}\left( {\bar{\varOmega }}\times [0,\infty )\right) ,\end{aligned}$$

(4.5.78)

$$\begin{aligned} v_{\varepsilon }\rightarrow v~~\text {in}~~C^0_{loc}\left( {\bar{\varOmega }}\times [0,\infty )\right) ,\end{aligned}$$

(4.5.79)

$$\begin{aligned} v_{\varepsilon }{\mathop {\rightharpoonup }\limits ^{*}} v~~\text {in}~~L^{\infty }((0,\infty );W^{1,r}(\varOmega ))~~\text {for each}~~r\in (1,\infty ),\end{aligned}$$

(4.5.80)

$$\begin{aligned} \nabla v_{\varepsilon }\rightarrow \nabla v~~\text {in}~~C^0_{loc}\left( {\bar{\varOmega }}\times [0,\infty )\right) ,\end{aligned}$$

(4.5.81)

$$\begin{aligned} u_{\varepsilon }\rightarrow u~~\text {in}~~C^0_{loc}\left( {\bar{\varOmega }}\times [0,\infty )\right) ,\end{aligned}$$

(4.5.82)

$$\begin{aligned} u_{\varepsilon }{\mathop {\rightharpoonup }\limits ^{*}}u ~~\text {in}~~L^{\infty }\left( \varOmega \times (0,\infty )\right) ,\end{aligned}$$

(4.5.83)

and

$$\begin{aligned} \nabla u_{\varepsilon }\rightharpoonup \nabla u~~\text {in}~~L^2_{loc}\left( {\bar{\varOmega }}\times [0,\infty )\right) .\end{aligned}$$

(4.5.84)

Furthermore, (n, c, v, u) solves (4.1.16), (4.1.22) and (4.1.23) globally in the sense of Definition 4.1.

Proof

Observing that

$$\int ^T_0\int _{\varOmega }|\nabla n^m_{\varepsilon }|^2=m^2\int ^T_0\int _{\varOmega }n^{2m-2}_{\varepsilon }|\nabla n_{\varepsilon }|^2\le m^2B_n\int ^T_0\int _{\varOmega }n^{2m-3}_{\varepsilon }|\nabla n_{\varepsilon }|^2$$

for each $T>0$ and all $\varepsilon \in (0,1),$ we thereby infer from (4.5.60) and (4.5.45) that actually $n^m_{\varepsilon }\in L^2_{loc}\left( [0,\infty );(W^{1,2}(\varOmega ))\right) .$ Thereupon, in line with the reasoning of Liu (2020, Lemma 7.2), the convergence claimed by (4.5.73)–(4.5.84) as well as the integral identities (4.5.3)–(4.5.6) are valid.

4.5.5 Asymptotic Behavior

Recalling (4.5.9) and (4.5.10), one can see that with some sufficiently small $\varepsilon _{*}\in (0,1)$ fulfilling

$$\begin{aligned} B_n\le \frac{1}{\varepsilon _{*}}\end{aligned}$$

(4.5.85)

(4.5.17) can be rewritten as

$$\int ^{\infty }_0\int _{\varOmega } n_{\varepsilon }c_{\varepsilon }\le C\quad \text {for all}~~\varepsilon \in (0,\varepsilon _{*}),$$

where $C>0.$ This in conjunction with (4.5.18), the convergence of $(n_{\varepsilon })_{\varepsilon \in (0,1)}$ and $(v_{\varepsilon })_{\varepsilon \in (0,1)}$ in Lemma 4.59 as well as the uniform boundedness property of $(n_{\varepsilon })_{\varepsilon \in (0,1)}$ implies the following stability of the spatial average of both n and v. The detailed reasoning thereof can be found in Liu (2020).

Lemma 4.60

Suppose that $\aleph \subset (0,\infty )$ is the null set provided by Lemma 4.59. Then we have

$$\begin{aligned} \int _{\varOmega } n (\cdot ,t)\rightarrow \left\{ \int _{\varOmega } n_0-\int _{\varOmega } v_0\right\} _{+}\quad \text {as}~~(0,\infty )\backslash \aleph \ni t\rightarrow \infty \end{aligned}$$

(4.5.86)

and

$$\begin{aligned} \int _{\varOmega } v (\cdot ,t)\rightarrow \left\{ \int _{\varOmega } v_0-\int _{\varOmega } n_0\right\} _{+}\quad \text {as}~~(0,\infty )\backslash \aleph \ni t\rightarrow \infty .\end{aligned}$$

(4.5.87)

Now, we are able to achieve the stability of both v and c as asserted by (4.1.28).

Lemma 4.61

Both v and c have the properties that

$$\begin{aligned} v\rightarrow v_{\infty }\quad \text {in}~~W^{1,\infty }(\varOmega )~~\text {as}~~ t\rightarrow \infty \end{aligned}$$

(4.5.88)

and

$$\begin{aligned} c\rightarrow v_{\infty }\quad \text {in}~~W^{1,\infty }(\varOmega )~~\text {as}~~ t\rightarrow \infty ,\end{aligned}$$

(4.5.89)

respectively, where $v_{\infty }=\frac{1}{|\varOmega |}\left\{ \int _{\varOmega }v_0-\int _{\varOmega }n_0\right\} _{+}.$

Proof

According to the arguments of Liu (2020, Lemmas 8.3–8.4), the convergence (4.5.81) together with (4.5.18) shows the uniform boundedness features of $\nabla v$ in $L^2(\varOmega \times (0,\infty ))$ by Fatou’s lemma, which along with the Poincaré inequality, (4.5.87) and the continuity of v implied by (4.5.79) entails the convergence $v\rightarrow v_{\infty }$ as $t\rightarrow \infty$ in the topology of $L^2(\varOmega ).$ In view of the embedding $C^{1+\nu }({\bar{\varOmega }})\hookrightarrow W^{1,\infty }(\varOmega )\hookrightarrow L^2(\varOmega )$ with the first one being compact, (4.5.88) follows from an Ehrling type interpolation argument relying on the Hölder regularity property of $\nabla v$ implied by (4.5.69). With the aid of (4.5.88), the convergence $c\rightarrow v_{\infty }$ as $t\rightarrow \infty$ in $L^2(\varOmega )$ can be derived from applications of a standard testing procedure along with the dominated convergence theorem to the second equation in (4.5.7) on the basis of (4.5.16), (4.5.76) and (4.5.79), based on which and the Hölder continuity of $\nabla c$ implied by (4.5.68), the convergence (4.5.89) is proved to be valid also from an Ehrling type lemma.

For the large time behavior of n,  we intend to divide the discussion into two situations, that are $\int _{\varOmega }n_0\le \int _{\varOmega }v_0$ and $\int _{\varOmega }n_0>\int _{\varOmega }v_0,$ where in the case when $\int _{\varOmega }n_0>\int _{\varOmega }v_0,$ a quasi-energy structure which resembles that constructed in Winkler (2018c) is essential to be analyzed for detecting the corresponding stability of n.

Lemma 4.62

With $\aleph \subset (0,\infty )$ as chosen in Lemma 4.59, for $\int _{\varOmega }n_0\le \int _{\varOmega }v_0,$ we have

$$\begin{aligned} n (\cdot ,t)\rightarrow n_{\infty }\quad \text {in}~~L^1(\varOmega )~~\text {as}~~(0,\infty )\backslash \aleph \ni t\rightarrow \infty ,\end{aligned}$$

(4.5.90)

while for $\int _{\varOmega }n_0>\int _{\varOmega }v_0,$ we have

$$\begin{aligned} n (\cdot ,t)\rightarrow n_{\infty }\quad \text {in}~~L^2(\varOmega )~~\text {as}~~(0,\infty )\backslash \aleph \ni t\rightarrow \infty ,\end{aligned}$$

(4.5.91)

where $n_{\infty }=\frac{1}{|\varOmega |}\left\{ \int _{\varOmega }n_0-\int _{\varOmega }v_0\right\} _{+}.$

Proof

If $\int _{\varOmega }n_0\le \int _{\varOmega }v_0,$ then clearly $n_{\infty }=0,$ whence (4.5.90) is an immediate consequence of (4.5.85). Whereas, if $\int _{\varOmega }n_0>\int _{\varOmega }v_0,$ in line with the reasoning of Liu (2020, Lemma 8.6), it is essential to firstly establish an inequality as follows, which shows the quantity $\int _{\varOmega }(n_{\varepsilon }-n_{\infty })^2$ remains small during a certain short time, that is for any fixed $t_{*}\ge 0$

$$\begin{aligned} \begin{aligned}\int _{\varOmega }\Big (n_{\varepsilon }(\cdot ,t)-n_{\infty }\Big )^2\le&C_1\cdot \left\{ \int _{\varOmega }\Big (n_{\varepsilon }(\cdot ,t_{*})-n_{\infty }\Big )^2+\int _{\varOmega }|\nabla c_{\varepsilon }(\cdot ,t_{*})|^2\right. \\&\left. +\int ^{t}_{t_{*}}\int _{\varOmega }v_{\varepsilon }n_{\varepsilon }+\sup _{s\in (t_{*},t_{*}+1)}\int _{\varOmega }|\nabla v_{\varepsilon }(\cdot ,s)|^2\right\} \end{aligned}\end{aligned}$$

(4.5.92)

for all $t\in (t_{*},t_{*}+1)$ and $\varepsilon \in (0,\varepsilon _{*})$ with some $C_1>0$ and $\varepsilon _{*}\in (0,1)$ satisfying (4.5.85), where $\int _{\varOmega }(n_{\varepsilon }(\cdot ,t_{*})-n_{\infty })^2$ can be verified to be arbitrarily small whenever $t_{*}$ is sufficiently large. Consequently, along with the decay properties of the last three integrals on the right-hand side of (4.5.92), as claimed by Lemma 4.50, (4.5.91) can be obtained.

Thanks to the bounds of n in $L^{\infty }(\varOmega )$ and the continuity implied by (4.5.75), the topologies in which n converges to $n_{\infty }$ as $t\rightarrow \infty$ as asserted by Lemma 4.62 can be further improved.

Lemma 4.63

each $p\ge 1,$

$$\begin{aligned} n (\cdot ,t)\rightarrow n_{\infty }\quad \text {in}~~L^p(\varOmega )~~\text {as}~~ t\rightarrow \infty \end{aligned}$$

(4.5.93)

holds.

Proof

As performed in the proof of Liu (2020, Corollary 8.7), the topology of the convergence claimed by (4.5.93) can be achieved by drawing on the Hölder inequality on the basis of the boundedness property of n in $L^{\infty }(\varOmega )$ as well as the stability of n provided by Lemma 4.62. Moreover, in light of the continuity implied by (4.5.75), the restriction that the convergence should be valid outside null sets of times as required by Lemma 4.62 can be removed. As a result, (4.5.93) follows.

The convergence of n and v in (4.5.93) and (4.5.88), respectively, enables us to derive the large time behavior of u from employing the variation-of-constants formula along with smoothing features of analytic semigroup, as demonstrated in the arguments of Liu (2020, Lemma 8.8).

Lemma 4.64

For u,  we have

$$\begin{aligned} u(\cdot ,t)\rightarrow 0\quad \text {in}~~L^{\infty }(\varOmega )~~\text {as}~~t\rightarrow \infty . \end{aligned}$$

(4.5.94)

Proof

Readers can find the detailed proof in Liu (2020).

Proof of Theorem 4.6. Theorem 4.6 follows from a collection of Lemmas 4.59, 4.61, 4.63 and 4.64.