Chemotaxis–Fluid System

Abstract

In the early 1970s, Keller and Segel proposed the cross-diffusion system to describe the phenomenon of spatial structures in biological system through chemical induced processes ([93, 94]).

1.1 Introduction

In the early 1970s, Keller and Segel proposed the cross-diffusion system to describe the phenomenon of spatial structures in biological system through chemical induced processes (Keller and Segel 1970, 1971a). In particular, they looked at situations where cells partially orient their movement along gradients of a signal secreted by themselves, or instead, cells direct their movement in response to a substance which they consume. A prototypical example of the former is the Dictyostelium discoideum colony, while the latter is an E. coli population. The model in which the biased migration is induced by the consumed nutrient is usually called chemotaxis–consumption system. Often such chemotactic movements take place in a fluid environment, and experimental findings and analytical studies have revealed the remarkable effects of chemotaxis–fluid interaction on the overall behavior of the respective chemotaxis systems, such as the prevention of blow-up and improvement of efficiency of mixing (Chertock et al. 2012; Kiselev and Ryzhik 2012a; Kiselev and Xu 2016; Lorz 2012; Tuval et al. 2005). It should be noted that the derivation of chemotaxis models interacting with a fluid can be obtained by asymptotic methods inspired by Hilbert’s sixth problem (Bellomo et al. 2016).

This chapter is concerned with a convective chemotaxis system for the oxygen-consuming and oxy-tactic bacteria, coupled with the incompressible Navier–Stokes equations. Section 1.3 is concerned with the following system

$$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\varDelta n^m -\nabla \cdot ( nS(x,n,c)\cdot \nabla c),&x\in \varOmega , t>0,\\&c_t+u\cdot \nabla c=\varDelta c-nc,&x\in \varOmega , t>0,\\&u_t+\nabla P=\varDelta u+n\nabla \phi ,&x\in \varOmega , t>0,\\&\nabla \cdot u=0,&x\in \varOmega , t>0,\\&\displaystyle {(\nabla n^m-nS(x,n,c)\cdot \nabla c)\cdot \nu =\partial _\nu c=0,u=0,}&x\in \partial \varOmega , t>0,\\&\displaystyle {n(x,0)=n_0(x),c(x,0)=c_0(x),u(x,0)=u_0(x),}&x\in \varOmega \end{aligned}\right. \end{aligned}$$

(1.1.1)

with $m > 0$, where $\varOmega$ is a bounded domain in $\mathbb {R}^{3}$, S(x, n, c) is a chemotactic sensitivity tensor satisfying

$$\begin{aligned} S\in C^2(\bar{\varOmega }\times [0,\infty )^2;\mathbb {R}^{3\times 3}) \end{aligned}$$

(1.1.2)

and

$$\begin{aligned} |S(x, n, c)|\le (1+n)^{-\alpha }S_0(c) ~~~~\text {for all}~~ (x, n, c)\in \varOmega \times [0,\infty )^2 \end{aligned}$$

(1.1.3)

with $\alpha \ge 0$ and some non-decreasing $S_0: [0,\infty )\rightarrow \mathbb {R}$.

In the two-dimensional analogue of (1.1.1), the condition of $m=1$, $\alpha =0$ is sufficient to ensure global existence of some generalized solution thereof (see Winkler 2018d), which eventually becomes smooth (Winkler 2021a). Nevertheless, a new difficulty arises in the analytical studies of the three-dimensional version of (1.1.1). It is well known that, compared with the case $m = 1$ (see Ke and Zheng 2019, Wang et al. 2018, Winkler 2018e), the nonlinear diffusion mechanism $m\ne 1$ may inhibit the occurrence of blow-up phenomena (see Tao and Winkler 2012b, Winkler 2013). Up to now, system (1.1.1) with nonlinear diffusion has been studied systematically. Indeed, in three space dimensions ($N=3$), many authors considered the global existence and boundedness of the solutions, and the restriction on m is weakened bit by bit. For example, when the chemotactic sensitivity function S(x, n, c) is scalar-value, in 2010, the range of m can be belong to $[\frac{7+\sqrt{217}}{12}, 2]$ (Francesco et al. 2010); in 2013, for locally bounded solution, m can be greater than $\frac{8}{7}$ (Tao and Winkler 2013); in 2018, the result is pushed to $m>\frac{9}{8}$ (Winkler 2018c). If we only consider the global existence of the solutions, rather than its boundedness, the value of m can be even smaller, such as $m\ge 1$ in Duan and Xiang (2014) and $m\ge \frac{2}{3}$ in Zhang and Li (2015a). When the chemotactic sensitivity function S(x, n, c) is tensor-value (S is a matrix), in 2015, Winkler (Winkler 2015b) established the uniform-in-time boundedness of global weak solutions in bounded and convex domains $\varOmega$ for $m>\frac{7}{6}$. Zheng (2022) extended the previous global boundedness result to $m>\frac{10}{9}.$ As an extension of this result, when S fulfills (1.1.3), the corresponding results are constantly updated. In 2017, it was shown in Wang and Li (2017) that $m\ge 1$ and $m+\alpha >\frac{7}{6}$ insures the global existence of bounded weak solution; in 2020, Wang (2020) extended the previous global boundedness result to $m +\frac{5}{4}\alpha>\frac{9}{8},\alpha >0$ and $m+\alpha >\frac{10}{9}.$ The first section shows how far the porous medium type diffusion of bacteria and saturation of tensor-valued sensitivity ensure the global boundedness of the weak solutions to (1.1.1) in the standard sense by the method different from those in Tao and Winkler (2013), Wang (2020), Winkler (2015b), Winkler (2018c).

In order to prepare a precise statement of our main results in these respects, let us assume that the initial data satisfy

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

(1.1.4)

where A denotes the Stokes operator with domain $D(A) {:=} W^{2,{2}}(\varOmega )\cap W^{1,{2}}_0(\varOmega )\cap L^{2}_{\sigma }(\varOmega )$, and $L^{2}_{\sigma }(\varOmega ) := \{\varphi \in L^{2}(\varOmega )|\nabla \cdot \varphi = 0\}$ (see Sohr 2001). As for the time-independent gravitational potential function $\phi$, we assume for simplicity that $\phi \in W^{2,\infty }(\varOmega )$.

Within this framework, our main result can be stated as follows (Zheng and Ke 2021):

Theorem 1.1

Let (1.1.4) hold and suppose that S satisfies (1.1.2)–(1.1.3). If $m+\alpha >\frac{10}{9}$ with $m >0$ and $\alpha \ge 0$, then there exists at least one global weak solution (in the sense of Definition 1.1 below) of problem (1.1.1). Also, this solution is bounded in $\varOmega \times (0,\infty )$ in the sense that for all $t > 0$

$$ \Vert n(\cdot , t)\Vert _{L^\infty (\varOmega )}+\Vert c(\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}+\Vert u(\cdot , t)\Vert _{L^{\infty }(\varOmega )}\le C $$

with some positive constant C independent of t. Moreover, c and u are continuous in $\bar{\varOmega }\times [0,\infty )$ and

$$ n\in C^0_{\omega -*}([0,\infty ); L^\infty (\varOmega )). $$

The proof of Theorem 1.1 focuses on the derivation of regularity estimates for the component $n_\varepsilon$ properly by means of a new bootstrap iteration in the case of $\frac{10}{9}<m+\alpha <\frac{3}{2}$, which seems to be quite different from those in Tao and Winkler (2013), Wang (2020), Winkler (2015b, 2018c), Zheng (2022). More precisely, based on the basic a priori estimates, we can establish the $L^{p}(\varOmega )$-estimates on $n_\varepsilon$ for some $p > \frac{3}{2}$ in the case $\frac{10}{9}<m+\alpha \le 2,\alpha > \frac{7}{18}$ or $m+ \alpha > 2$ by using some carefully analysis. Whereas for $\frac{10}{9}< m+ \alpha \le 2$ and smaller $\alpha \in [0,\frac{7}{18}]$, the derivation of the $L^{p*}(\varOmega )$-estimates on $n_\varepsilon$ with some $p*>\frac{3}{2}$ needs a new iteration. In fact, on the basis of the spatio-temporal estimate $\int _{t}^{t+1}\int _{\varOmega }\frac{n_{\varepsilon }}{c_{\varepsilon }}|\nabla c_{\varepsilon }|^2$ provided by the quasi-energy functional, one can establish the boundedness of $n_\varepsilon$ in $L^{p_1}(\varOmega )$ (see Lemma 1.20) and $L^{p_n}(\varOmega )$ (see Lemma 1.23), where $p_1= \frac{16}{3}(m+\alpha )^2-\frac{25}{3}(m+\alpha )+4+\frac{1}{3}\alpha [4(m+\alpha )-1]$ and $p_n=\frac{2}{3}p_{n}^2+\frac{2}{3}(4m-5+3\alpha )p_n+(2m+2\alpha -3)(m-1)+1$. Based on the $L^{p_n}$-boundedness of $n_\varepsilon$, one can then archive the uniform bounds of $n_\varepsilon$ in $L^p(\varOmega )$ for any $p>1$. With the aid of a standard Morse-type technique and the maximal Sobolev regularity, one can derive the boundedness of $n_\varepsilon ,$ $\nabla c_\varepsilon$ and $u_\varepsilon$ in $L^\infty (\varOmega )$, inter alia the further regularity properties thereof which seem necessary to obtain the global weak solution to system (1.1.1).

In the second part of this chapter, we are concerned with the chemotaxis–consumption system coupled with the incompressible Navier–Stokes equations

$$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\triangle n-\chi \nabla \cdot \left( \displaystyle \frac{n}{c}\nabla c\right) +f(n),&x\in \varOmega , t>0,\\&c_t+u\cdot \nabla c=\triangle c-nc,&x\in \varOmega , t>0,\\&u_t+ (u\cdot \nabla ) u=\varDelta u+\nabla P+n\nabla \phi ,&x\in \varOmega , t>0,\\&\nabla \cdot u=0,&x\in \varOmega , t>0, \end{aligned}\right. \end{aligned}$$

(1.1.5)

describing the biological population density n, the chemical signal concentration c, the incompressible fluid velocity u and the associated pressure P of the fluid flow in the physical domain $\varOmega \subset \mathbb {R}^N$. It is assumed that n and c diffuse randomly as well as are transported by the fluid, with a buoyancy effect on n through the presence of a given gravitational potential $\phi$. Further, it is assumed that the chemotactic stimulus is perceived in accordance with the Weber–Fechner Law (Short et al. 2010; Wang 2013; Winkler 2019b) which states that subjective sensation is proportional to the logarithm of the stimulus intensity, in other words, the population n partially direct their movement toward increasing concentrations of the chemical nutrient c that they consume with the logarithmic sensitivity. In addition, on the considered time scales of cell migration, we allow for population growth to take place, through the term $f(n)=rn-\mu n^2$ with the effective growth rate $r\in \mathbb {R}$, which accounts for the mortality or population renewal, and strength of the overcrowding effect $\mu >0$; we note that $r=0$ is allowed and has indeed been argued for in certain models (Hillen and Painter 2009; Kiselev and Ryzhik 2012a).

The system (1.1.5) appears to generate interesting, nontrivial dynamics. However, to the best of our knowledge, no analytical result is available yet which rigorously describes the qualitative behavior of such solutions. This may be due to the circumstance that (1.1.5) joins two subsystems which are far from being fully understood even when decoupled from each other. Indeed, (1.1.5) contains the Navier–Stokes equations which themselves do not admit a complete existence and regularity theory (Wiegner 1999).

At the same time, by setting $u\equiv 0$ in (1.1.5), we arrive at the following chemotaxis–consumption model

$$\begin{aligned} \left\{ \begin{aligned}&n_t=\triangle n-\chi \nabla \cdot \left( \frac{n}{c}\nabla c\right) ,\\&c_t=\triangle c-nc, \end{aligned}\right. \end{aligned}$$

(1.1.6)

where population growth has been ignored, which was introduced by Keller and Segel (1971a) to describe the collective behavior of the bacteria E. coli set in one end of a capillary tube featuring a gradient of nutrient concentration observed in the celebrated experiment of Adler (1966). Later, this model was also employed to describe the dynamical interactions between vascular endothelial cells and vascular endothelial growth factor (VEGF) during the initiation of tumor angiogenesis (see Corrias et al. 2003; Levine et al. 2000). It has already been demonstrated that the logarithmic sensitivity featured in (1.1.6) renders a significant degree of complexity in the system; in particular, it plays an indispensable role in generating wave-like solutions without any type of cell kinetics (Hillen and Painter 2009; Keller and Segel 1970; Rosen 1978; Schwetlick 2003; Wang 2013), which is a prominent feature in the Fisher equation (Kolmogorov et al. 1937).

In comparison with (1.1.6), the related chemotaxis system

$$\begin{aligned} \left\{ \begin{aligned}&n_t=\triangle n-\chi \nabla \cdot (\frac{n}{c}\nabla c)+f(n),\\&c_t=\triangle c-c+n, \end{aligned}\right. \end{aligned}$$

(1.1.7)

where the chemical signal c is actively secreted by the bacteria rather than consumed (see Bellomo et al. 2015; Hillen and Painter 2009), has been more extensively studied. It is observed that the chemical signal production mechanism in the $c-$equation inhibits the tendency of c to take on small values, and thereby the singularity in the sensitivity function is mitigated. Accordingly, for such higher dimensional systems with reasonably smooth but arbitrarily large data, the global existence of bounded smooth solutions can be achieved. Indeed, global existence and boundedness of classical solutions to (1.1.7) without source terms is guaranteed if $\chi \in (0,\sqrt{\frac{2}{N}})$ (Fujie 2015; Winkler 2011a), or if $N=2, \chi \in (0,\chi _0)$ with some $\chi _0>1.015$ (Lankeit 2016b), while certain generalized solutions have been constructed for general $\chi >0$ in the two-dimensional radially symmetric case (Stinner and Winkler 2011; Winkler 2011a). Moreover, without any symmetry hypothesis, Winkler and Lankeit established the global solvability of generalized solutions for the cases $\chi <\infty , N=2$; $\chi <\sqrt{8}, N=3$; and $\chi <\frac{N}{N-2}, N\ge 4$ (Lankeit and Winkler 2017).

Furthermore, in accordance with known results for the classical Keller–Segel chemotaxis model (see Lankeit 2015; Winkler 2010a, 2014a for example), the presence of the logistic source term $f(n)=n(r-\mu n)$ in (1.1.7) can inhibit the tendency toward explosions of cells at least under some restrictions on certain parameters. Indeed, it is known that (1.1.7) with $N = 2$ possesses a global classical solution (n, c) for any $r\in \mathbb {R}, \chi , \mu >0$, and (n, c) is globally bounded if $r>\frac{\chi ^2}{4}$ for $0 < \chi \le 2$ or $r > \chi -1$ for $\chi >2$ (Zhao and Zheng 2017). Moreover, (n, c) exponentially converges to $(\frac{r}{\mu },\frac{r}{\mu })$ in $L^\infty (\varOmega )$ provided that $\mu >0$ is sufficiently large (Zheng et al. 2018). As for the higher dimensional cases ($N\ge 2$), the global very weak solution of (1.1.7) with $f(n)=r n-\mu n^k$ is constructed when $k,\chi$ and r fulfill a certain condition. In addition, when $N=2$ or 3, this solution is global bounded provided $\frac{r}{\mu }$ and the initial data $\Vert n_0\Vert _{L^2}, \Vert \nabla c_0\Vert _{L^4}$ are suitably small (Zhao and Zheng 2019).

In contrast to (1.1.7), system (1.1.6) is more challenging due to the combination of the consumption of c with the singular chemotaxis sensitivity of n. Intuitively, the absorption mechanism in the $c-$equation of (1.1.6), which induces the preference for small values of c, considerably intensifies the destabilizing potential of singular sensitivity in the $n-$equation. Up to now, it seems that only limited results on global classical solvability in the spatial two-dimensional case are available. In fact, only recently have certain global generalized solutions to (1.1.6) been constructed for general initial data in Lankeit and Lankeit (2019b), Winkler (2016a), Winkler (2018a), whereas with respect to global classical solvability, it has only been shown for some small initial data (see Wang et al. 2016; Winkler 2016c). In particular, Winkler (2016c) showed that the global classical solutions to (1.1.6) in bounded convex two-dimensional domains exist and converge to the homogeneous steady state under an essentially explicit smallness condition on $n_0$ in $L\log L(\varOmega )$ and $\nabla \ln c_0$ in $L^2(\varOmega )$. We would, however, like to note that numerous variants of (1.1.6), such as those involving nonlinear diffusion, logistic-type cell kinetics and saturating signal production (Ding and Zhao 2018; Jia and Yang 2019; Lankeit and Lankeit 2019a; Lankeit 2017; Winkler 2022; Lankeit and Viglialoro 2020; Liu 2018; Viglialoro 2019; Zhao and Zheng 2018), have been studied. For example, the authors of Zhao and Zheng (2018) proved that the particular version of (1.1.6) by adding $f(n)=rn-\mu n^{k}~ (r>0, \mu>0, k>1)$ into the n-equation admits a global classical solution (n, c) in the bounded domain $\varOmega \subset \mathbb {R}^N$ if $k>1+\frac{N}{2}$, and in the two-dimensional setting, $(n,c, \frac{|\nabla c|}{c})\rightarrow ((\frac{r}{\mu })^{\frac{1}{k-1}},0,0)$ for sufficiently large $\mu$. In particular, it is shown in the recent paper Lankeit and Lankeit (2019a) that (1.1.6) with logistic source $f(n)=r n-\mu n^2~ (r\in \mathbb {R}, \mu >0)$ possesses a unique global classical solution if $0<\chi <\sqrt{\frac{2}{N}}$, $\mu >\frac{N-2}{2N}$, and a globally bounded solution only in one dimension for any $\chi>0, \mu >0$. Also, the author of Wang (2019) showed that if $\mu >\mu _0$ with some $\mu _0=\mu _0(\varOmega ,\chi )>0$ then the corresponding classical solution is globally bounded, and $(n,c, \frac{|\nabla c|}{c})\rightarrow (\frac{r_+}{\mu },\lambda ,0)$ with $\lambda \in [0,\frac{1}{|\varOmega |}\int _\varOmega c_0)$ in $(L^\infty (\varOmega ))^3$ as $t\rightarrow \infty$. Of course, this leaves open the possibility of blow-up of solutions when $\mu$ is positive but small. Anyhow, it has been shown in Winkler (2017a) that when $\mu >0$ is suitably small, the strongly destablizating action of chemotactic cross-diffusion may lead to the occurrence of solutions which attain possibly finite but arbitrarily large values.

Coming back to our chemotaxis–consumption–fluid model (1.1.5), as we have already pointed out, very little seems to be known regarding the qualitative behavior of solutions (Black 2018; Black et al. 2018, 2019). In fact, we are aware of one result only which is concerned with the asymptotic behavior and eventual regularity of solutions to the Stokes variant of (1.1.5). Namely, it is shown in Black (2018) that for small initial mass $\int _\varOmega n_0$, the corresponding system upon neglection of $u\cdot \nabla u$ and f(n) in (1.1.5) possesses at least one global generalized solutions, which will become smooth after some waiting time and stabilize toward the steady state $(\frac{1}{|\varOmega |}\int _\varOmega n_0,0,0)$ with respect to the topology of $(L^\infty (\varOmega ))^3$. Since the presence of the fluid interaction does not have any regularizing effect on the large time behavior, it is expected that instead of the small restriction on the initial data, the quadratic degradation may have a substantial regularizing effect on the dynamic behavior of solutions to (1.1.5).

The second part of this chapter focuses on the asymptotic profile in time of solutions to (1.1.5) in the two-dimensional case. In order to state our main results, we shall impose on (1.1.5) the boundary conditions

$$\begin{aligned} \nabla n\cdot \nu =\nabla c\cdot \nu =0\quad \text {and}~~u=0 ~~ \text {for}~~ x\in \partial \varOmega , \end{aligned}$$

(1.1.8)

and initial conditions

$$\begin{aligned} n(x,0)=n_0(x),~ c(x,0)=c_0(x), ~ u(x,0)=u_0(x)~ ~\text {for}~~x\in \varOmega . \end{aligned}$$

(1.1.9)

Throughout this part, it is assumed that

$$\begin{aligned} \left\{ \begin{aligned}&n_0\in C^0(\bar{\varOmega }), ~n_0\ge 0~ \text {and}~ n_0\not \equiv 0 ~ \text {in}~ \varOmega ,\\&c_0\in W^{1,\infty }(\varOmega ), c_0>0 ~ \text {in}~ \bar{\varOmega }~~\text {as well as} \\&u_0\in D(A^{\beta })\;\text {for all}\;\beta \in (\frac{1}{2},1) \end{aligned}\right. \end{aligned}$$

(1.1.10)

with A denoting the Stokes operator $A{=}-\mathscr {P}\varDelta$ with domain $D(A){:=}W^{2,2}(\varOmega ; \mathbb {R}^2)\cap W^{1,2}_0(\varOmega ; \mathbb {R}^2)\cap L_\sigma ^2(\varOmega )$, where $L_\sigma ^2(\varOmega ):=\{\varphi \in L^2(\varOmega ; \mathbb {R}^2)|\nabla \cdot \varphi =0\}$ and $\mathscr {P}$ stands for the Helmholtz projection of $L^2(\varOmega )$ onto $L_\sigma ^2(\varOmega )$.

Within this framework, by straightforward adaptation of arguments in Lankeit and Lankeit (2019a) with only some necessary modifications, one can see that the problem (1.1.5), (1.1.8), (1.1.9) admits a global classical solution (n, c, u, P) whenever $\chi \in (0,1)$, $r\in \mathbb {R}$ and $\mu >0$, which is unique up to addition of constants in the pressure variable P, and satisfies $n > 0$, $c > 0$ in $\varOmega \times [0,\infty )$. The first of our main results is concerned with the global boundedness of the solution as well as its asymptotic behavior (Pang et al. 2021).

Theorem 1.2

Let $f(n)=rn-\mu n^2$, $r\in \mathbb {R}$, $\mu >0$ and $\phi \in W^{2,\infty }(\varOmega )$, and suppose that $(n_0,c_0,u_0)$ satisfy (1.1.10). If (n, c, u, P) denotes the corresponding global classical solution to (1.1.5), (1.1.8), (1.1.9), then there exists a value $\mu _0=\mu _0(\varOmega ,\chi , r)\ge 0$ with $\mu _0(\varOmega ,\chi , 0)= 0$ such that whenever $\mu >\mu _0$, (n, c, u) is global bounded,

$$\Vert n(\cdot ,t)-\frac{r_+}{\mu }\Vert _{L^\infty (\varOmega )}\rightarrow 0, ~~ \Vert \frac{\nabla c}{c}(\cdot ,t)\Vert _{L^\infty (\varOmega )} \rightarrow 0,~~ \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )} \rightarrow 0 ~~ $$

and when $r>0$, $\Vert c(\cdot ,t) \Vert _{L^\infty (\varOmega )} \rightarrow 0$ as $t \rightarrow \infty$.

As indicated in the above discussion, we need to introduce new ideas to show how the regularizing effect of the quadratic degradation in the chemotaxis–fluid model (1.1.5) can counterbalance the strongly destabilizating action of chemotactic cross-diffusion caused by the combination of the consumption of c with the singular chemotaxis sensitivity of n. Specifically, we develop the conditional energy functional method in Winkler (2016c) to show the global boundedness of solutions in the case of $r>0$, in which the key point is to verify that

$$\begin{aligned} \mathscr {F}(n,w):= \int _\varOmega H(n) +\frac{\chi }{2}\displaystyle \int _\varOmega |\nabla w|^2, ~~w:= -\ln (\frac{c}{\Vert c_0\Vert _{L^\infty (\varOmega )}}) \end{aligned}$$

(1.1.11)

with $H(s):=s\ln \frac{\mu s}{er}+ \frac{r}{\mu }$ constitutes an energy functional in the sense that $\mathscr {F}(n,w)$ is non-increasing in time whenever $\mu$ is appropriately large relative to r (see Lemma 1.41). Indeed, from (1.4.29), one can obtain the global bound of $\int _\varOmega n|\ln n| dx$ and $\int _\varOmega |\nabla w|^2 dx$, which then serves as a starting point to derive the uniform bound of $\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ via the Neumann heat semigroup estimates. Furthermore, by making appropriate use of the dissipative information expressed in (1.4.29), we can establish the convergence result asserted in Theorem 1.2. It is noted that compared to that of the case $r>0,\mu >0$, the proof of Theorem 1.2 in the case of $r\le 0, \mu >0$ involves a more delicate analysis. In fact, unlike in the case $r=\mu =0$ or $r>0,\mu >0$, (1.1.5) with $r\le 0,\mu >0$ seems to lack the favorable structure that facilitates such conditional energy-type inequalities. Taking full advantage of the decay information on n in $L^1-$norm expressed in (1.4.2), our approach toward Theorem 1.2 is to construct the quantity

$$\begin{aligned} \mathscr {F}(n,w):= \int _\varOmega n(\ln n+a) +\frac{\chi }{2}\displaystyle \int _\varOmega |\nabla w|^2 \end{aligned}$$

(1.1.12)

with parameter $a>0$ determined below (see (1.4.64)). Unlike in the case of $r>0$, $\mathscr {F}(n,w)$ does not enjoy monotonicity property, it however satisfies a favorable non-homogeneous differential inequality (1.4.71) in the sense that it can provide us a priori information on solution such as the global bound of $\int _\varOmega n|\ln n| dx$ and $\int _\varOmega |\nabla w|^2 dx$ (see Lemma 1.43), as well as $\displaystyle \lim _{t\rightarrow \infty }\int _\varOmega |\nabla w(\cdot ,t)|^2=0$ (see (1.4.77)).

As an important step to understand the model (1.1.5) more comprehensively, we shall consider the convergence rate of its classical solutions in the form of the following result:

Theorem 1.3

Let the assumptions of Theorem 1.2 hold and $r>0$. Then one can find $\mu _*(\chi , \varOmega , r)>0$ such that if $\mu >\mu _*(\chi , \varOmega , r)$, the classical solution of (1.1.5), (1.1.8), (1.1.9) presented in Theorem 1.2 satisfies

$$\Vert n(\cdot ,t)-\frac{r}{\mu }\Vert _{L^\infty (\varOmega )}\rightarrow 0, ~~ \Vert c(\cdot ,t) \Vert _{L^\infty (\varOmega )} \rightarrow 0,~~~~ \Vert u(\cdot ,t) \Vert _{L^\infty (\varOmega )} \rightarrow 0 $$

as well as $\Vert \frac{\nabla c}{c}(\cdot ,t)\Vert _{L^p(\varOmega )} \rightarrow 0$ for all $p>1$ exponentially as $t\rightarrow \infty$.

This implies that suitably large $\mu$ relative to r enforces asymptotic stability of the corresponding constant equilibria of (1.1.5); however, the optimal lower bound on $\frac{\mu }{r}$ seems yet lacking. The main ingredient of our approach toward Theorem 1.3 involves a so-called self-map-type reasoning. More precisely, making use of the convergence properties of $(n,\frac{|\nabla c|}{c})$ asserted in Theorem 1.3, we prove by a self-map-type reasoning that whenever $\mu$ is suitably large compared with r,

$$(n(\cdot ,t)-\frac{r}{\mu }, c(\cdot ,t),u)\longrightarrow (0,0,0) ~~ \hbox {and}~~\frac{|\nabla c|}{c}(\cdot ,t)\longrightarrow 0 $$

in $(L^\infty (\varOmega ))^3$ and $L^6(\varOmega )$ exponentially as $t\rightarrow \infty$, respectively (see Lemma 1.45).

As aforementioned, the limit case $r=0$ becomes relevant in several applications. In this limiting situation, the total cell population can readily be seen to decay in the large time limit (cf. Lemma 1.36 below). As a consequence, we can obtain the decay properties of solutions, namely that the decay on n in $L^1$ actually occurs in $L^\infty$, and also for c. More precisely, our result reads as follows:

Theorem 1.4

Let the assumptions of Theorem 1.2 hold and $r=0$. Then the classical solution of (1.1.5), (1.1.8), (1.1.9) from Theorem 1.2 satisfies $(n,c,\frac{|\nabla c|}{c},u)\longrightarrow (0,0,0,0)$ in $(L^\infty (\varOmega ))^4$ algebraically as $t\rightarrow \infty$.

The result indicates that structure generating dynamics in the spatially two-dimensional version of (1.1.5), (1.1.8) and (1.1.9), if at all, occur on intermediate time scales rather than in the sense of a stable large time pattern formation process. Apparently, it leaves open the questions whether the more colorful large time behavior can appear in the three-dimensional version of (1.1.5).

The approach toward Theorem 1.4 uses an alternative method, which, at its core, is based on the argument that the $L^\infty$-norm of n can be controlled from above by appropriate multiples of $\frac{1}{t+1}$. This results from a suitable variation-of-constants representation of n, by which and in view of the decay information on $|\nabla w|$ in $L^\infty (\varOmega )$, the $L^1$ decay information on u from (1.4.2) can be turned into the $L^\infty$-norm of n (see Lemma 1.46). As a consequence, by comparison argument, we have a pointwise upper estimate for w as well as a lower estimate for v (see Lemma 1.47). Using $L^p-L^q$ estimates for the Neumann heat semigroup $(e^{t\varDelta })_{t>0}$, we then successively show that $\Vert \nabla w\Vert _{L^\infty }$ and $\Vert n\Vert _{L^\infty (\varOmega )}$ can be controlled by appropriate multiples of $\frac{1}{t+1}$ from above and below, respectively (see Lemma 1.48). These a priori estimates allow us to get the pointwise lower estimate for w as well as the upper estimate for c, which complement the lower bound for c previously obtained, and thereby prove that c actually decays algebraically.

1.2 Preliminaries

Firstly let us recall the important $L^p-L^q$ estimates for the Neumann heat semigroup $(e^{t\varDelta })_{t>0}$ on bounded domains, which plays an important role not only in Chap. 1, but also in Chaps. 3, 4 and 6.

Lemma 1.1

(Lemma 1.3 of Winkler 2010 and Lemma 2.1 of Cao 2015) Let $(e^{t\varDelta })_{t>0}$ denote the Neumann heat semigroup in the domain $\varOmega$ and $\lambda _1>0$ denote the first nonzero eigenvalue of $-\varDelta$ in $\varOmega \subset \mathbb R^N$ under the Neumann boundary condition. There exists $c_i$, $i=1,2,3,4$, such that for all $t>0$,

(i) If $1\le q\le p\le \infty$, then for all $\omega \in L^q(\varOmega )$ with $\int _\varOmega \omega =0$,

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

(ii) If $1\le q\le p\le \infty$, then for all $\omega \in L^q(\varOmega )$,

$$\Vert \nabla e^{t\varDelta }\omega \Vert _{L^p(\varOmega )}\le c_2\left( 1+t^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right) e^{-\lambda _1t}\Vert \omega \Vert _{L^q(\varOmega )};$$

(iii) If $2\le q\le p\le \infty$, then for all $\omega \in W^{1,q}(\varOmega )$,

$$\Vert \nabla e^{t\varDelta }\omega \Vert _{L^p(\varOmega )}\le c_3\left( 1+t^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right) e^{-\lambda _1t}\Vert \nabla \omega \Vert _{L^q(\varOmega )};$$

(iv) If $1\le q\le p<\infty$ or $1<q<\infty$ and $p=\infty$, then for all $\omega \in (L^q(\varOmega ))^N$,

$$\Vert e^{t\varDelta }\nabla \cdot \omega \Vert _{L^p(\varOmega )}\le c_4\left( 1+t^{-\frac{1}{2}-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}\right) e^{-\lambda _1t}\Vert \omega \Vert _{L^q(\varOmega )}.$$

In order to obtain the solution of system (1.1.1) through a suitable approximation procedure, we follow the well-established approaches to regularize both the chemotactic sensitivity and nonlinear diffusion in the first equation in (1.1.1) (see Cao and Lankeit 2016; Li et al. 2015; Winkler 2015a, b; Ke and Zheng 2019). Let $(\rho _\varepsilon )_{\varepsilon \in (0,1)} \in C^\infty _0 (\varOmega )$ be a family of standard cut-off functions, which satisfying $0\le \rho _\varepsilon \le 1$ in $\varOmega$ and $\rho _\varepsilon \nearrow 1$ in $\varOmega$ as $\varepsilon \searrow 0$, and $\chi _\varepsilon \in C^\infty _0 ([0,\infty ))$ satisfying $0\le \chi _\varepsilon \le 1$ in $[0,\infty )$ and $\chi _\varepsilon \nearrow 1$ as $\varepsilon \searrow 0$. Define

$$ S_\varepsilon (x, n, c) := \rho _\varepsilon (x)\chi _\varepsilon (n)S(x, n, c),~~ x\in \bar{\varOmega },~~n\ge 0,~~c\ge 0~~ $$

for $\varepsilon \in (0, 1)$, which implies that $S_\varepsilon (x, n, c) = 0$ on $\partial \varOmega$. As an approximation function of the sensitivity tensor S, $S_\varepsilon$ also satisfies the condition (1.1.3), that is,

$$\begin{aligned} |S_\varepsilon (x, n, c)|\le (1+n)^{-\alpha } S_0(c) ~~~~\text {for all}~~ (x, n, c)\in \varOmega \times [0,\infty )^2. \end{aligned}$$

(1.2.1)

The regularized problem of (1.1.1) can be presented as follows

$$\begin{aligned} \left\{ \begin{aligned}&n_{\varepsilon t}+u_{\varepsilon }\cdot \nabla n_{\varepsilon }=\varDelta (n_{\varepsilon }+\varepsilon )^m -\nabla \cdot (n_{\varepsilon }F_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_{\varepsilon }),&x\in \varOmega , t>0, \\&c_{\varepsilon t}+u_{\varepsilon }\cdot \nabla c_{\varepsilon }=\varDelta c_{\varepsilon }-n_{\varepsilon }c_{\varepsilon },&x\in \varOmega , t>0, \\&u_{\varepsilon t}+\nabla P_{\varepsilon }=\varDelta u_{\varepsilon }+n_{\varepsilon }\nabla \phi ,&x\in \varOmega , t>0, \\&\nabla \cdot u_{\varepsilon }=0,&x\in \varOmega , t>0, \\&\nabla n_{\varepsilon }\cdot \nu =\nabla c_{\varepsilon }\cdot \nu =0,u_{\varepsilon }=0,&x\in \partial \varOmega , t>0, \\&n_{\varepsilon }(x,0)=n_0(x),c_{\varepsilon }(x,0)=c_0(x),u_{\varepsilon }(x,0)=u_0(x),&x\in \varOmega , \end{aligned} \right. \end{aligned}$$

(1.2.2)

where $F_{\varepsilon }(s)=\frac{1}{1+\varepsilon s}~~\text {for}~~s \ge 0$.

Let us recall the local well-posedness of (1.2.2).

Lemma 1.2

(Winkler 2012, 2015b) Let $\varOmega \subseteq \mathbb {R}^3$ be a bounded domain with smooth boundary. Suppose that (1.1.2)–(1.1.3) hold. Assume that the initial data $(n_0,c_0,u_0)$ fulfills (1.1.4). Then for each $\varepsilon \in (0, 1)$, there exist functions

$$\begin{aligned} \left\{ \begin{aligned}&n_{\varepsilon }\in C^0(\bar{\varOmega }\times [0,\infty ))\cap C^{2,1}(\bar{\varOmega }\times (0,\infty )),\\&c_{\varepsilon }{}\in C^0(\bar{\varOmega }\times [0,\infty ))\cap C^{2,1}(\bar{\varOmega }\times (0,\infty ))\cap _{q>3}C^0([0,\infty );W^{1,q}(\varOmega )),\\&u_{\varepsilon }{}\in C^0(\bar{\varOmega }\times [0,\infty ))\cap C^{2,1}(\bar{\varOmega }\times (0,\infty )),\\&P_{\varepsilon }{}\in C^{1,0}(\bar{\varOmega }\times (0,\infty )),\\ \end{aligned}\right. \end{aligned}$$

(1.2.3)

such that $(n_{\varepsilon }, c_{\varepsilon }, u_{\varepsilon }, P_{\varepsilon })$ solves (1.2.2) classically in $\varOmega \times (0,\infty )$, and such that $n_\varepsilon \ge 0$ and $c_\varepsilon >0$ in $\bar{\varOmega }\times (0,\infty )$.

The following lemma reveals the relationship between the regularity of $u_\varepsilon$ and $n_\varepsilon$.

Lemma 1.3

(Winkler 2015b; Zheng 2022, 2019) Let $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon ,P_\varepsilon )$ be the solution of (1.2.2) in $\varOmega \times (0,T)$ as well as $p \in [1,+\infty )$ and $q \in [1,+\infty )$, such that

$$ \left\{ \begin{aligned} q<\frac{3p}{3-p}~~\hbox {if}~~p\le 3, \\ q\le \infty ~~\hbox {if}~~p>3. \end{aligned} \right. $$

Then for all $K > 0$, there exists $C = C(p,q, K)$ such that if $\Vert n_{\varepsilon }(\cdot , t)\Vert _{L^p(\varOmega )}\le K$ for all $t\in (0, T)$, then $\Vert D u_{\varepsilon }(\cdot , t)\Vert _{L^q(\varOmega )}\le C$ for all $t\in (0, T)$.

The following lemmas will be used in the sequel.

Lemma 1.4

Let $T>0$, $\tau \in (0,T)$, $A>0,\alpha >0$ and $B>0$, and suppose that $y:[0,T)\rightarrow [0,\infty )$ is absolutely continuous fulfilling $y'(t)+Ay^\alpha (t)\le h(t)$ for a.e. $t\in (0,T)$ with some nonnegative function $h\in L^1_{loc}([0, T))$ satisfying $\int _{t}^{t+\tau }h(s)ds\le B~~\text {for all}~~t\in (0,T-\tau ).$ Then

$$\begin{aligned} y(t)\le \max \left\{ y_0+B,\frac{1}{\tau ^{\frac{1}{\alpha }}}(\frac{B}{A})^{\frac{1}{\alpha }}+2B\right\} ~~\text {for all}~~t\in (0,T). \end{aligned}$$

(1.2.4)

For its elementary proof, we refer to Lemma 3.4 of Stinner et al. (2014) where the particular case $\tau =\alpha = 1$ is detailed.

As a crucial tool for analyzing the key term $\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}$ below, we will use the following inequality established by Lemma 2.2.4 in Lankeit (2016a).

Lemma 1.5

(Lankeit 2016a) There are $C_0> 0$ and $\mu _0 > 0$ such that every positive $w\in C^2(\bar{\varOmega })$ fulfilling $\nabla w\cdot \nu = 0$ on $\partial \varOmega$ satisfies

$$\begin{aligned} \begin{aligned}&-2\int _{\varOmega }\frac{|\varDelta w|^2 }{ w}+\int _{\varOmega }\frac{|\nabla w|^2\varDelta w }{ w^2} \\ \le&-\mu _0\int _{\varOmega } w|D^2\ln w|^2-\mu _0\int _{\varOmega }\frac{|\nabla w|^4}{ w^3}+C_0\int _{\varOmega } w. \end{aligned} \end{aligned}$$

(1.2.5)

Now, we display an important auxiliary interpolation lemma in Winkler (2015b), Zheng and Wang (2017).

Lemma 1.6

(Winkler 2015b; Zheng and Wang 2017) Let $q\ge 1$,

$$\begin{aligned} \lambda \in [2q+2,4q+1] \end{aligned}$$

(1.2.6)

and $\varOmega \subset \mathbb {R}^3$ be a bounded domain with smooth boundary. Then there exists $C > 0$ such that for all $\varphi \in C^2(\bar{\varOmega })$ fulfilling $\varphi \cdot \frac{\partial \varphi }{\partial \nu }= 0$ on $\partial \varOmega$, we have

$$\begin{aligned} \Vert \nabla \varphi \Vert _{L^\lambda (\varOmega )}\le C\Vert |\nabla \varphi |^{q-1}D^2\varphi \Vert _{L^2(\varOmega )}^{\frac{2(\lambda -3)}{(2q-1)\lambda }} \Vert \varphi \Vert _{L^\infty (\varOmega )}^{\frac{6q-\lambda }{(2q-1)\lambda }}+C\Vert \varphi \Vert _{L^\infty (\varOmega )}. \end{aligned}$$

(1.2.7)

As an application of Lemma 1.6, (1.2.10) immediately leads to

Lemma 1.7

Let $\beta \in [1,\infty )$. Then there exists a positive constant $\lambda _{0,\beta }$ such that

$$\begin{aligned} \Vert \nabla c_\varepsilon \Vert _{L^{2\beta +2}(\varOmega )}^{2\beta +2}\le \lambda _{0,\beta }(\Vert |\nabla c_\varepsilon |^{\beta -1}D^2c_\varepsilon \Vert _{L^2(\varOmega )}^{2}+1). \end{aligned}$$

(1.2.8)

The basic boundedness information of solutions to (1.2.2) is stated as follows.

Lemma 1.8

The solution $(n_{\varepsilon }, c_{\varepsilon }, u_{\varepsilon }, P_{\varepsilon })$ of (1.2.2) satisfies

$$\begin{aligned} \Vert n_{\varepsilon }(\cdot ,t)\Vert _{L^1(\varOmega )}=\Vert n_0\Vert _{L^1(\varOmega )}~~~\text {for all}~~ t>0 \end{aligned}$$

(1.2.9)

and

$$\begin{aligned} \Vert c_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \Vert c_0\Vert _{L^\infty (\varOmega )}~~~\text {for all}~~ t>0. \end{aligned}$$

(1.2.10)

Proof

The identity (1.2.9) directly follows by integrating the first equation in (1.2.2). Moreover (1.2.10) is readily derived by applying the maximum principle to the second equation.

The following Gagliardo–Nirenberg inequality will be used several times in Sect. 1.4.

Lemma 1.9

Let $\varOmega \subset \mathbb {R}^2$ be a bounded Lipschitz domain. Then i) there is $C> 0$ such that $\Vert \nabla \varphi \Vert ^4_{L^4(\varOmega )}\le C\Vert \varDelta \varphi \Vert ^2_{L^2(\varOmega )}\Vert \nabla \varphi \Vert ^2_{L^2(\varOmega )}$for all $\varphi \in W^{2,2}(\varOmega )$ fulfilling $\frac{\partial \varphi }{\partial \nu }|_{\partial \varOmega }=0$; (ii) there is $C> 0$ such that $\Vert \varphi \Vert ^3_{L^3(\varOmega )}\le C\Vert \varphi \Vert ^2_{W^{1,2}(\varOmega )}\Vert \varphi \Vert _{L^1(\varOmega )}$ for all $\varphi \in W^{1,2}(\varOmega )$.

1.3 Global Boundedness of Solution to a Chemotaxis–Fluid System with Nonlinear Diffusion

1.3.1 A Quasi-energy Functional

Since some first regularity properties beyond those from Lemma 1.8 can be obtained by making use of a quasi-energy functional. Indeed it is a starting point of the derivation of further estimates for solutions to the approximate problems (1.2.2).

Lemma 1.10

For any $\varepsilon \in (0,1)$, the solution $(n_{\varepsilon }, c_{\varepsilon }, u_{\varepsilon }, P_{\varepsilon })$ of (1.2.2) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }} + \mu _0 \int _{\varOmega }c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2+\frac{\mu _0}{2}\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3} +\int _{\varOmega }\frac{n_{\varepsilon }|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}}\\ \le&\displaystyle { 2\int _{\varOmega }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }| + \frac{2\Vert c_0\Vert _{L^\infty (\varOmega )}}{\mu _0} \int _{\varOmega }|\nabla u_{\varepsilon }|^2 +C~~~\text {for all}~~ t>0} \end{aligned} \end{aligned}$$

(1.3.1)

for some $C> 0$, where $\mu _0$ is the same as (1.2.5).

Proof

Thanks to $c_\varepsilon > 0$, we integrate by parts and deduce from $c_\varepsilon$-equation in (1.2.2) that

$$\begin{aligned} \begin{aligned} \displaystyle \frac{d}{dt}\displaystyle \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }} =&\displaystyle {-2\int _{\varOmega }\frac{\varDelta c_{\varepsilon } c_{\varepsilon t}}{c_{\varepsilon }} +\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2c_{\varepsilon t}}{c_{\varepsilon }^2}}\\ =&\displaystyle {-2\int _{\varOmega }\frac{|\varDelta c_{\varepsilon }|^2 }{c_{\varepsilon }} +2\int _{\varOmega }\frac{\varDelta c_{\varepsilon } n_{\varepsilon } c_{\varepsilon }}{c_{\varepsilon }} +2\int _{\varOmega }\frac{\varDelta c_{\varepsilon }}{c_{\varepsilon }}u_{\varepsilon }\cdot \nabla c_{\varepsilon }}\\&+\displaystyle {\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2\varDelta c_{\varepsilon } }{c_{\varepsilon }^2} -\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2n_{\varepsilon } c_{\varepsilon } }{c_{\varepsilon }^2} -\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2u_{\varepsilon }\cdot \nabla c_{\varepsilon } }{c_{\varepsilon }^2}}\\ =&\displaystyle {-2\int _{\varOmega }\frac{|\varDelta c_{\varepsilon }|^2 }{c_{\varepsilon }} +2\int _{\varOmega }\varDelta c_{\varepsilon }n_{\varepsilon } +2\int _{\varOmega }\frac{\varDelta c_{\varepsilon }}{c_{\varepsilon }}u_{\varepsilon }\cdot \nabla c_{\varepsilon }}\\&+\displaystyle {\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2\varDelta c_{\varepsilon } }{c_{\varepsilon }^2} -\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2n_{\varepsilon }}{c_{\varepsilon }} -\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2u_{\varepsilon }\cdot \nabla c_{\varepsilon } }{c_{\varepsilon }^2}}. \end{aligned} \end{aligned}$$

(1.3.2)

Together with (1.2.10), an application of Lemma 1.5 yields that for some positive constants $\mu _0$ and $C(\mu _0)$, it has

$$ \begin{aligned}&-2\int _{\varOmega }\frac{|\varDelta c_{\varepsilon }|^2 }{c_{\varepsilon }} +\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2\varDelta c_{\varepsilon } }{c_{\varepsilon }^2} \\ \le&-\mu _0\int _{\varOmega }(c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2+ \frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3})+C(\mu _0)\Vert c_{0}\Vert _{L^\infty (\varOmega )}|\varOmega |~~~\text {for}~~ t>0. \end{aligned} $$

In addition, integrating by parts again, we have and

$$ \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }^2}u_{\varepsilon }\cdot \nabla c_{\varepsilon } =2\int _{\varOmega }\frac{1}{c_{\varepsilon }}u_{\varepsilon }\cdot D^2c_{\varepsilon }\cdot \nabla c_{\varepsilon }~~\text {for all}~~ t>0. $$

So combining the above two inequalities, we get

$$ \begin{aligned}&2\int _{\varOmega }\displaystyle \frac{\varDelta c_{\varepsilon }}{c_{\varepsilon }}(u_{\varepsilon }\cdot \nabla c_{\varepsilon }) -\displaystyle \int _{\varOmega }\displaystyle \frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }^2}u_{\varepsilon }\cdot \nabla c_{\varepsilon } \\ \le&2\int _{\varOmega } \frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}|\nabla u_{\varepsilon }| \le \frac{\mu _0}{2}\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3}+ \frac{2\Vert c_0\Vert _{L^\infty (\varOmega )}}{\mu _0} \int _{\varOmega }|\nabla u_{\varepsilon }|^2. \end{aligned} $$

Therefore inequality (1.3.1) readily results from above inequalities.

In order to deal with the term $\int _{\varOmega }|\nabla u_{\varepsilon }|^2$ on the right of (1.3.1), we recall the following standard energy inequality for the fluid component of solutions of (1.2.2).

Lemma 1.11

Let $m+\alpha >\frac{2}{3}$. Then for any $\eta \in (0,1)$, there exists $C(\eta ) > 0$ such that

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }{|u_{\varepsilon }|^2}+\int _{\varOmega }{|\nabla u_{\varepsilon }|^2} \le \eta \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2+C(\eta ) ~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.3)

Proof

Testing the third equation in (1.2.2) by $u_\varepsilon$ and using $\nabla \cdot u_{\varepsilon }=0$, we get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _{\varOmega }{|u_{\varepsilon }|^2}+\int _{\varOmega }{|\nabla u_{\varepsilon }|^2} = \int _{\varOmega }n_{\varepsilon }u_{\varepsilon }\cdot \nabla \phi ~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.4)

By the Young inequality and the continuity of the embedding $W^{1,2}(\varOmega )\hookrightarrow L^6(\varOmega )$, we obtain that there is $C_1 > 0$ such that

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\varOmega }n_{\varepsilon }u_{\varepsilon }\cdot \nabla \phi \le&\displaystyle {\Vert \nabla \phi \Vert _{L^\infty (\varOmega )}\Vert n_{\varepsilon }\Vert _{L^{\frac{6}{5}}(\varOmega )}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}}\\ \le&\displaystyle {C_1\Vert n_{\varepsilon }+\varepsilon \Vert _{L^{\frac{6}{5}}(\varOmega )}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}~~\text {for all}~~ t>0}. \end{aligned} \end{aligned}$$

(1.3.5)

Further, by the Gagliardo–Nirenberg inequality, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \Vert n_{\varepsilon }+\varepsilon \Vert _{L^{\frac{6}{5}}(\varOmega )}\\ \le&C_2\displaystyle {(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{2}(\varOmega )}^{\frac{1}{3(m+\alpha )-1}}\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{\frac{2}{m+\alpha }}(\varOmega )}^{\frac{2}{m+\alpha }-\frac{1}{3(m+\alpha )-1}}} + \Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{\frac{2}{m+\alpha }}(\varOmega )}^{\frac{2}{m+\alpha }}) \\ \le&\displaystyle {C_3(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{2}(\varOmega )}^{\frac{1}{3(m+\alpha )-1}}+1)~~\text {for all}~~ t>0,} \end{aligned} \end{aligned}$$

(1.3.6)

for some $C_2>0$ and $C_3>0$ independent of $\varepsilon$. Combining (1.3.6) with (1.3.5) and noticing $m+\alpha >\frac{2}{3}$, we can see that for any $\eta \in (0,1)$,

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{\varOmega }n_{\varepsilon }u_{\varepsilon }\cdot \nabla \phi \\ \le&\displaystyle {\frac{1}{2}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2 +C_4(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{2}(\varOmega )}^{\frac{2}{3(m+\alpha )-1}}+1)}\\ \le&\displaystyle {\frac{1}{2}\Vert \nabla u_{\varepsilon }\Vert _{L^{2}(\varOmega )}^2 +\eta \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2+C(\eta )~~\text {for all}~~ t>0}. \end{aligned} \end{aligned}$$

(1.3.7)

This together with (1.3.4) arrives at (1.3.3).

Now, we turn to analyze $\int _{\varOmega }(n_{\varepsilon }+\varepsilon ) \ln (n_{\varepsilon }+\varepsilon )$ or $\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )}$, which contributes to absorbing $\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2$ on the right-hand side of (1.3.3).

Lemma 1.12

The solution $(n_{\varepsilon }, c_{\varepsilon }, u_{\varepsilon }, P_{\varepsilon })$ of (1.2.2) satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\frac{d}{dt}\int _{\varOmega }(n_\varepsilon +\varepsilon ) \ln (n_\varepsilon +\varepsilon ) +m\int _\varOmega (n_\varepsilon +\varepsilon )^{m-2}|\nabla n_\varepsilon |^2 \\ \le&C_S\int _\varOmega |\nabla n_\varepsilon ||\nabla c_{\varepsilon }|~~\qquad \text {if}~~\alpha =0, \\&\frac{1}{{\alpha (1+\alpha )}}\frac{d}{dt}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )} +m\int _{\varOmega }({n_{\varepsilon }}+\varepsilon )^{m+\alpha -2} |\nabla n_{\varepsilon }|^2 \\ \le&C_S\int _\varOmega {|\nabla } {n}_{\varepsilon }||\nabla c_{\varepsilon }|~~\qquad \text {if}~~\alpha >0 \end{aligned} \right. \end{aligned}$$

(1.3.8)

for all $t>0$, where $C_S=\displaystyle \sup _{0\le s\le \Vert c_{0}\Vert _{L^\infty (\varOmega )}}S_0(s)$.

Proof

The proof of the lemma is given separately for two cases.

(1) For the case $\alpha =0$. Integration by parts, we deduce from $n_\varepsilon$-equation as well as $\nabla \cdot u_\varepsilon = 0$ and (1.2.1) that

$$\begin{aligned}&\displaystyle \frac{d}{dt}\displaystyle \int _{\varOmega }(n_{\varepsilon }+\varepsilon ) \ln (n_{\varepsilon }+\varepsilon ) \nonumber \\ =&\displaystyle {\int _{\varOmega }\varDelta (n_{\varepsilon }+\varepsilon )^m\ln (n_{\varepsilon }+\varepsilon )- \int _{\varOmega }\ln (n_{\varepsilon }+\varepsilon )\nabla \cdot (n_{\varepsilon }F_{\varepsilon } (n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_{\varepsilon })} \nonumber \\&\displaystyle {-\int _{\varOmega }\ln (n_{\varepsilon }+\varepsilon )u_{\varepsilon }\cdot \nabla n_{\varepsilon }} \\ \le&\displaystyle {-m\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m-2}|\nabla n_{\varepsilon }|^2+ \int _{\varOmega }S_0(c_{\varepsilon })|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|} \nonumber \\ \le&\displaystyle {-m\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m-2}|\nabla n_{\varepsilon }|^2+C_S \int _{\varOmega }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|}. \nonumber \end{aligned}$$

(1.3.9)

(2) For the case $\alpha >0$. Multiplying the first equation in (1.2.2) by $(n_\varepsilon +\varepsilon )^{\alpha }$, and noticing the hypothesis (1.1.3), we then have

$$\begin{aligned}&\displaystyle {\frac{1}{{1+\alpha }}\frac{d}{dt}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )}+ m{\alpha }\int _{\varOmega }({n_{\varepsilon }}+\varepsilon )^{m+\alpha -2} |\nabla n_{\varepsilon }|^2} \nonumber \\ =&\displaystyle {{\alpha }\int _\varOmega ({n}_{\varepsilon }+\varepsilon )^{\alpha -1}{n}_\varepsilon {\nabla } {n}_{\varepsilon }\cdot (F_{\varepsilon }(n_{\varepsilon })S_{\varepsilon }(x,n_{\varepsilon },c_{\varepsilon }) \cdot \nabla c_{\varepsilon })} \\ \le&\displaystyle {{\alpha }\int _\varOmega ({n}_{\varepsilon }+\varepsilon )^{\alpha } (1+n_\varepsilon )^{-\alpha }S_0(c_\varepsilon ){|\nabla } {n}_{\varepsilon }||\nabla c_{\varepsilon }|} \nonumber \\ \le&\displaystyle {{\alpha }C_S\int _\varOmega {|\nabla } {n}_{\varepsilon }||\nabla c_{\varepsilon }|~~\text {for all}~~ t>0.} \nonumber \end{aligned}$$

(1.3.10)

Hence, (1.3.8) readily follows from (1.3.9) and (1.3.10).

Remark 1.1

Note that when S(x, n, c) is scalar-value, one can make use of the corresponding flavor thereof to neutralize $2\int _{\varOmega }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|$ on the right-hand side of (1.3.1) and $C_S\int _{\varOmega } |\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|$ on the right side of (1.3.8).

In the sequel we shall derive an energy-type inequality under the assumption $\frac{10}{9}< m+\alpha \le 2$, from which the regularity of solutions of (1.2.2) beyond that of Lemma 1.8 is achieved.

Lemma 1.13

Let $\frac{10}{9}< m+\alpha \le 2$ and S satisfy (1.1.2)–(1.1.3). Suppose that (1.1.4) hold. Then there exists $C>0$ independent of $\varepsilon$ such that the solution of (1.2.2) satisfies

$$\begin{aligned} \displaystyle {\displaystyle \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}+\displaystyle \int _{\varOmega }|u_{\varepsilon }|^2\le C~~\text {for all}~~ t>0,} \end{aligned}$$

(1.3.11)

$$\begin{aligned} \left\{ \begin{aligned}&\displaystyle \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{1+\alpha }\le C~~~~\text {if}~~ \alpha >0,\\&\displaystyle \int _{\varOmega }(n_{\varepsilon }+\varepsilon )\ln (n_{\varepsilon }+\varepsilon )\le C~~\text {if}~~ \alpha =0, \end{aligned} \right. \end{aligned}$$

(1.3.12)

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{t}^{t+1}\int _{\varOmega }(\frac{n_{\varepsilon }}{c_{\varepsilon }}|\nabla c_{\varepsilon }|^2 +\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3}+(n_\varepsilon +\varepsilon )^{m+\alpha +\frac{2}{3}} + (n_{\varepsilon }+{\varepsilon })^{m+\alpha -2} |\nabla {n_{\varepsilon }}|^2)\le C,}\\ \end{aligned} \end{aligned}$$

(1.3.13)

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{t}^{t+1}\int _{\varOmega } |\nabla {u_{\varepsilon }}|^2\le C}\\ \end{aligned} \end{aligned}$$

(1.3.14)

as well as

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{t}^{t+1}\int _{\varOmega }c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2\le C.}\\ \end{aligned} \end{aligned}$$

(1.3.15)

Proof

Adding an suitable multiples of the inequalities in Lemmas 1.10–1.12, one can conclude that there exist positive constants $c_i, (i=1,2,3),$ such that

$$\begin{aligned}&\frac{d}{dt}\left( \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }} +\int _{\varOmega }(n_{\varepsilon }+\varepsilon ) \ln (n_{\varepsilon }+\varepsilon ) +k_1\int _{\varOmega }|u_{\varepsilon }|^2\right) +k_2 \int _{\varOmega }|\nabla u_{\varepsilon }|^2 \nonumber \\&+k_2 \varOmega c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2 +\int _{\varOmega }\frac{n_{\varepsilon }|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }} \\&+k_2 \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3}+ k_2 \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m-2}|\nabla n_{\varepsilon }|^2 \nonumber \\&\le \displaystyle {k_3 \int _{\varOmega }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }| + k_3~~~\text {for all}~~ t>0~~~\text {when}~~~\alpha =0} \nonumber \end{aligned}$$

(1.3.16)

and

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}\left( \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }} +\frac{1}{{\alpha (1+\alpha )}}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )}+k_1 \int _{\varOmega }|u_{\varepsilon }|^2\right) } \\&\displaystyle {+k_2 \int _{\varOmega }|\nabla u_{\varepsilon }|^2 +k_2 \int _{\varOmega }c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2+\int _{\varOmega }\frac{n_{\varepsilon }|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}}\\&\displaystyle {+k_2\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3} +k_2 \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2}\\ \le&\displaystyle {k_3 \int _{\varOmega }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }| + k_3~~~\text {for all}~~ t>0~~~\text {when}~~~\alpha >0.} \end{aligned} \end{aligned}$$

(1.3.17)

Next, we will estimate $\int _{\varOmega }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|$ by the Gagliardo–Nirenberg inequality along with the basic priori information provided by Lemma 1.8. Indeed, making use of (1.2.10) and the Young inequality, we thereby find $k_5> 0$ such that

$$\begin{aligned} \begin{aligned}&k_3\displaystyle \int _{\varOmega }|\nabla n_{\varepsilon }||\nabla c_{\varepsilon }|\\ \le&\displaystyle { \frac{k_2}{2} \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3} } +\displaystyle {\frac{k_2}{4}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^{2} +k_5\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{4-2m-2\alpha }}. \end{aligned} \end{aligned}$$

(1.3.18)

Therefore for $\alpha >0$, we insert (1.3.18) into (1.3.17) to get

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}\left( \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }} +\frac{1}{{\alpha (1+\alpha )}}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )} +k_1 \int _{\varOmega }|u_{\varepsilon }|^2\right) }\\&\displaystyle {+k_2 \int _{\varOmega }|\nabla u_{\varepsilon }|^2 +k_2 \int _{\varOmega }c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2+\int _{\varOmega }\frac{n_{\varepsilon }|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}}\\&\displaystyle {+\frac{k_2}{2}\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3} +\frac{3k_2}{4} \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2}\\ \le&k_5\displaystyle \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{4-2m-2\alpha } + k_3~~~\text {for all}~~ t>0. \end{aligned} \end{aligned}$$

(1.3.19)

Next, we deal with $\displaystyle \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{4-2m-2\alpha }$ separately for two cases. Indeed, in the case $\frac{10}{9}<m+\alpha <\frac{3}{2}$, by the Gagliardo–Nirenberg inequality, we get

$$\begin{aligned}&k_5\displaystyle \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{4-2m-2\alpha } \nonumber \\ =&\displaystyle {k_5\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{\frac{2(4-2m-2\alpha )}{m+\alpha }}(\varOmega )}^{\frac{2(4-2m-2\alpha )}{m+\alpha }}} \nonumber \\ \le&k_6\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{2}(\varOmega )}^{\frac{2(3-2m-2\alpha )}{3(m+\alpha )-1}}\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{\frac{2}{m+\alpha }}(\varOmega )}^{\frac{2(4-2m-2\alpha )}{m+\alpha }-\frac{2(3-2m-2\alpha )}{3(m+\alpha )-1}} \\&+\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{\frac{2}{m+\alpha }}(\varOmega )}^{\frac{2(4-2m-2\alpha )}{m+\alpha }} \nonumber \\ =&\displaystyle {k_7(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+\alpha }{2}}\Vert _{L^{2}(\varOmega )}^{\frac{6(3-2m-2\alpha )}{3(m+\alpha )-1}}+1)~~\text {for all}~~ t>0,} \nonumber \end{aligned}$$

(1.3.20)

where $k_6$ and $k_7$ are positive constants. Hence, if $\frac{10}{9}<m+\alpha <\frac{3}{2}$, we have $\frac{6(3-2m-2\alpha )}{3(m+\alpha )-1}\in (0,2),$ and then get

$$\begin{aligned} \begin{aligned}&\displaystyle {k_5\displaystyle \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{4-2m-2\alpha }\le \frac{k_2}{4}\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2+k_8~~\text {for all}~~ t>0}\\ \end{aligned} \end{aligned}$$

(1.3.21)

with some $k_8>0$ by the Young inequality. While in the case $\frac{3}{2}\le m+\alpha \le 2$, we have $4-2m-2\alpha \in (0,1)$ and thereby immediately get

$$\begin{aligned} k_5\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{4-2m-2\alpha } \le k_5 \int _{\varOmega }n_{\varepsilon }+k_9 \end{aligned}$$

(1.3.22)

with some $k_9>0$. Therefore, (1.3.19) together with (1.3.20)–(1.3.22) leads to

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}\left( \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }} +\frac{1}{{\alpha (1+\alpha )}}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )} +k_1 \int _{\varOmega }|u_{\varepsilon }|^2\right) }\\&\displaystyle {+k_2 \int _{\varOmega }|\nabla u_{\varepsilon }|^2 +k_2 \int _{\varOmega }c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2+\int _{\varOmega }\frac{n_{\varepsilon }|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}}\\&\displaystyle {+\frac{k_2}{2}\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3} +\frac{k_2}{2} \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2}\\&\le k_{10}~~~\text {for all}~~ t>0. \end{aligned} \end{aligned}$$

(1.3.23)

Since $m+\alpha >\frac{10}{9}$, we utilize the Gagliardo–Nirenberg inequality to see that there exists a positive constant $k_{11}$ such that

$$ \begin{aligned}&\displaystyle \frac{1}{{\alpha (1+\alpha )}}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )} \\ \le&\displaystyle {k_{11} \left( \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2\right) ^{\frac{3\alpha }{3(m+\alpha )-1}}+k_{11} ~~ \text {for all}~~ t>0.} \end{aligned} $$

Hence recalling (1.2.10) and according to the Poincaré inequality, we can see that for all $t>0$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}+\frac{1}{{\alpha (1+\alpha )}}\Vert {n_{\varepsilon }} +\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )} +k_1\int _{\varOmega }|u_{\varepsilon }|^2}\\ \le&\displaystyle {k_{12}\left( \int _{\varOmega }|\nabla u_{\varepsilon }|^2+\int _{\varOmega } \frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3}+\int _{\varOmega }(n_{\varepsilon } +\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2\right) ^\zeta +k_{12}} \end{aligned} \end{aligned}$$

(1.3.24)

with some $k_{12}>0$ and $\zeta =\max \{{\frac{3\alpha }{3(m+\alpha )-1}},1\}$. Thus, we infer from (1.3.23) and (1.3.24) that there exist $k_{13}> 0$ and $k_{14}> 0$ such that for all $\varepsilon \in (0, 1)$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}\left( \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}+\frac{1}{{\alpha (1+\alpha )}}\Vert {n_{\varepsilon }} +\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )} +\frac{8}{\mu _0}\Vert c_0\Vert _{L^\infty (\varOmega )}\int _{\varOmega }|u_{\varepsilon }|^2\right) }\\&+\displaystyle {k_{13}\left( \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}+\frac{1}{{\alpha (1+\alpha )}}\Vert {n_{\varepsilon }} +\varepsilon \Vert ^{{{1+\alpha }}}_{L^{{1+\alpha }}(\varOmega )} +\frac{8}{\mu _0}\Vert c_0\Vert _{L^\infty (\varOmega )}\int _{\varOmega }|u_{\varepsilon }|^2\right) ^{\frac{1}{\zeta }}}\\&\displaystyle {+k_{13}\left( \int _{\varOmega }|\nabla u_{\varepsilon }|^2+\int _{\varOmega }c_{\varepsilon }|D^2\ln c_{\varepsilon }|^2 +\int _{\varOmega }\frac{n_{\varepsilon }|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}\right) }\\&\displaystyle {+k_{13}\left( \int _{\varOmega }\frac{|\nabla c_{\varepsilon }|^4}{c_{\varepsilon }^3} +\frac{5m\alpha }{8}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+\alpha -2}|\nabla n_{\varepsilon }|^2\right) }\\&\le \displaystyle { k_{14}~~~\text {for all}~~ t>0~~~\text {if}~~~\alpha >0} \end{aligned} \end{aligned}$$

(1.3.25)

which along with Lemma 1.4, implies that (1.3.11)–(1.3.12) are valid. Further, (1.3.13)–(1.3.15) result from integrating the inequality (1.3.25). The proof for the case $\alpha =0$ can be proved similarly, and is thus omitted here.

1.3.2 $L^\infty ((0,\infty );L^p(\varOmega ))$ Estimate of $n_\varepsilon$ for Some $p > \frac{3}{2}$

The further regularity properties of solutions can be obtained by means of a bootstrap iteration in the case of $\frac{10}{9}<m+\alpha <\frac{3}{2}$. In this direction, we first shall make use of results in Lemma 1.13 to improve the regularities, in particular for $n_{\varepsilon }$.

Lemma 1.14

Let $p>1$. Then the solution $(n_{\varepsilon }, c_{\varepsilon }, u_{\varepsilon }, P_{\varepsilon })$ of (1.2.2) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{p}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ \frac{{m}(p-1)}{2}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{m+p-3} |\nabla n_{\varepsilon }|^2}\\ \le&\displaystyle {\frac{(p-1)C_S^2}{2m}\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p+1-m}(1+n_{\varepsilon })^{-2\alpha }|\nabla c_{\varepsilon }|^2~~\text {for all}~~ t>0.}\\ \end{aligned} \end{aligned}$$

(1.3.26)

Proof

Multiplying the first equation in (1.2.2) by $({n_{\varepsilon }}+\varepsilon )^{p-1}$, using $\nabla \cdot u_\varepsilon =0$ as well as (1.1.3), we get

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{p}}\frac{d}{dt}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ m(p-1)\int _{\varOmega }({n_{\varepsilon }}+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2}\\ =&\displaystyle {(p-1)\int _\varOmega ({n}_{\varepsilon }+\varepsilon )^{p-2}{n}_\varepsilon {\nabla } {n}_{\varepsilon } \cdot (F_{\varepsilon }(n_{\varepsilon })S_{\varepsilon }(x,n_{\varepsilon },c_{\varepsilon }) \cdot \nabla c_{\varepsilon })}\\ \le&\displaystyle {(p-1)\int _\varOmega ({n}_{\varepsilon } +\varepsilon )^{p-1}(1+n_\varepsilon )^{-\alpha }S_0(c_\varepsilon ){|\nabla } {n}_{\varepsilon }||\nabla c_{\varepsilon }|}\\ \le&\displaystyle {(p-1)C_S\int _\varOmega ({n}_{\varepsilon } +\varepsilon )^{p-1}(1+n_\varepsilon )^{-\alpha }{|\nabla } {n}_{\varepsilon }||\nabla c_{\varepsilon }|~~\text {for all}~~ t>0}. \end{aligned} \end{aligned}$$

(1.3.27)

Hence (1.3.26) follows from (1.3.27) and Young’s inequality.

As a consequence of Lemma 1.13, we have

Lemma 1.15

Under the assumptions of Lemma 1.13, there exists a positive constant C independent of $\varepsilon$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega }|\nabla c_{\varepsilon }|^2\le C~~\text {for all}~~ t>0.}\\ \end{aligned} \end{aligned}$$

(1.3.28)

Proof

Noticing that $|\nabla c_{\varepsilon }|^2\le \displaystyle \frac{|\nabla c_{\varepsilon }|^2}{c_{\varepsilon }}\Vert c_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )}$, (1.3.28) results from (1.3.11) and (1.2.10).

Combining Lemma 1.14 and estimate (1.3.28) immediately leads to

Lemma 1.16

Let $\frac{10}{9}< m+\alpha \le 2$, S satisfy (1.1.2)–(1.1.3) and $(n_{\varepsilon }, c_{\varepsilon }, u_{\varepsilon }, P_{\varepsilon })$ be the solution of (1.2.2). Then there exists $C>0$ independent of $\varepsilon$ such that

$$\begin{aligned} \displaystyle { \sup _{t\in (0,\infty )}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+2\alpha } + \sup _{t\in (0,\infty )}\int _{t}^{t+1}\int _{\varOmega } (n_{\varepsilon }+\varepsilon )^{2m+2\alpha -3} |\nabla {n_{\varepsilon }}|^2 \le C} \end{aligned}$$

(1.3.29)

for all $t>0$.

Proof

Taking $p = m+2\alpha$ in (1.3.26), we get

$$\begin{aligned} \begin{aligned}&\frac{1}{{m+2\alpha }}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{m+2\alpha }}}_{L^{{m+2\alpha }}(\varOmega )}+ \frac{{m}(m+2\alpha -1)}{2}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -3} |\nabla n_{\varepsilon }|^2 \\ \le&k_1\int _\varOmega (n_{\varepsilon }+\varepsilon )^{1+2\alpha }(1+n_{\varepsilon })^{-2\alpha }|\nabla c_{\varepsilon }|^2 \\ \le&\Vert c_0\Vert _{L^\infty (\varOmega )} k_1\int _\varOmega \frac{n_{\varepsilon }}{c_{\varepsilon }}|\nabla c_{\varepsilon }|^2+k_1\int _\varOmega |\nabla c_{\varepsilon }|^2 \end{aligned} \end{aligned}$$

(1.3.30)

for some positive constant $k_1>0$. Now, applying the Gagliardo–Nirenberg inequality and (1.2.9), one can find constants $k_2>0$, $k_3>0$ and $k_4>0$ independent of $\varepsilon \in (0,1)$ such that

$$\begin{aligned}&\displaystyle \int _{\varOmega }(n_\varepsilon +\varepsilon )^{m+2\alpha } \nonumber \\ =&\displaystyle {\Vert (n_\varepsilon +\varepsilon )^{\frac{2m+2\alpha -1}{2}}\Vert ^{\frac{2 (m+2\alpha )}{2m+2\alpha -1}}_{L^{ \frac{2 (m+2\alpha )}{2m+2\alpha -1}}(\varOmega )}} \nonumber \\ \le&c_{2}\left( \Vert \nabla (n_\varepsilon +\varepsilon )^{\frac{2m+2\alpha -1}{2}} \Vert _{L^2(\varOmega )}^{\frac{3m+6\alpha -3}{3m+3\alpha -2}}\Vert (n_\varepsilon +\varepsilon )^{\frac{2m+2\alpha -1}{2}} \Vert _{L^\frac{2}{2m+2\alpha -1}(\varOmega )}^{\frac{2 (m+2\alpha )}{2m+2\alpha -1}-\frac{3m+6\alpha -3}{3m+3\alpha -2}}\right. \nonumber \\&\left. +\Vert (n_\varepsilon +\varepsilon )^{\frac{2m+2\alpha -1}{2}}\Vert _{L^\frac{2}{2m+2\alpha -1}(\varOmega )}^{\frac{2 (m+2\alpha )}{2m+2\alpha -1}}\right) \nonumber \\ \le&\displaystyle {c_{3}(\Vert \nabla (n_\varepsilon +\varepsilon )^{\frac{2m+2\alpha -1}{2}}\Vert _{L^{2}(\varOmega )}^{\frac{3m+6\alpha -3}{3m+3\alpha -2}}+1)} \nonumber \\ \le&\displaystyle {\frac{{m}(m-1)}{(2m-1)^2}\Vert \nabla (n_\varepsilon +\varepsilon )^{\frac{2m+2\alpha -1}{2}}\Vert _{L^{2}(\varOmega )}^{2}+k_4.} \nonumber \end{aligned}$$

Inserting the above inequality into (1.3.30), one then has

$$\begin{aligned} \begin{aligned}&\frac{1}{{m+2\alpha }}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{m+2\alpha }}}_{L^{{m+2\alpha }}(\varOmega )}+ \frac{{m}(m+2\alpha -1)}{4}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -3} |\nabla n_{\varepsilon }|^2 \\&+\int _{\varOmega }(n_\varepsilon +\varepsilon )^{m+2\alpha } \\ \le&\Vert c_0\Vert _{L^\infty (\varOmega )}\displaystyle k_1\int _\varOmega \frac{n_{\varepsilon }}{c_{\varepsilon }}|\nabla c_{\varepsilon }|^2+k_1\int _\varOmega |\nabla c_{\varepsilon }|^2+k_4. \end{aligned} \end{aligned}$$

As the application of Lemma 1.4, this together with (1.3.28) and (1.3.13) then arrives at (1.3.29).

According to Lemma 1.3, the bound of $L^p(\varOmega )$ for $Du_\varepsilon$ can be suitably enlarge upon the result of Lemma 1.16 in asserting the following.

Lemma 1.17

Let $\frac{10}{9}< m+\alpha \le \frac{3}{2}$. Then for $r<\frac{3(m+\alpha )}{3-(m+\alpha )}$, there exists $K:= K(r, m)$ such that

$$\begin{aligned} \Vert D u_{\varepsilon }(\cdot , t)\Vert _{L^r(\varOmega )}\le K~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.31)

Proof

In light of (1.3.29), (1.3.31) is the consequence of an application of Lemma 1.3 with $p=m+\alpha$.

In order to obtain the further regularity of $c_\varepsilon$, one can establish the time evolution of $\nabla c_\varepsilon$ in $L^{{2\beta }}(\varOmega )$, similar to that of Lemma 3.6 in Winkler (2015b).

Lemma 1.18

For any $\beta > 1$, the solution of (1.2.2) satisfies

$$\begin{aligned} \begin{aligned}&\frac{1}{2\beta } \frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{2\beta } _{L^{2\beta }(\varOmega )} +\frac{2(\beta -1)}{\beta ^2} \int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{\beta }\right| ^2 \\&+\frac{1}{2} \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta -2} |D^2c_{\varepsilon }|^2+ \int _{\varOmega }n_{\varepsilon } |\nabla c_{\varepsilon }|^{2\beta } \\ \le&\displaystyle { -\int _\varOmega c_{\varepsilon } |\nabla c_{\varepsilon }|^{2\beta -2}\nabla n_{\varepsilon }\cdot \nabla c_{\varepsilon } +\int _\varOmega |Du_{\varepsilon }| |\nabla c_{\varepsilon }|^{2\beta }}+C \end{aligned} \end{aligned}$$

(1.3.32)

for all $t>0$, where $C>0$ is a positive constant independent of $\varepsilon$.

Proof

Noticing the boundedness of $\Vert \nabla c_{\varepsilon }(\cdot ,t)\Vert _{L^2(\varOmega )}$ obtained in Lemma 1.15, and applying the arguments as those in the proof (3.10) of Ishida et al. (2014) (see also Wang and Xiang 2016; Zheng 2016, 2017a), one can find a positive constant $k_1$ such that

$$ \displaystyle {\int _{\partial \varOmega }\frac{\partial |\nabla c_{\varepsilon }|^2}{\partial \nu } |\nabla c_{\varepsilon }|^{2\beta -2} \le \frac{(\beta -1)}{{\beta ^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{\beta }\right| ^2+k_1.} $$

Hence by pursuing quite a similar strategy in the proof of Lemma 3.6 in Winkler (2015b), one can derive (1.3.32).

Now, we address the question how far the regularity information such as provided by Lemma 1.17 is convenient to estimate the term $\int _\varOmega |Du_{\varepsilon }| |\nabla c_{\varepsilon }|^{2\beta }$ on the right of (1.3.32).

Lemma 1.19

Let $r>\frac{3}{2}$ and $\beta \in [r-1,\frac{r-1}{(4-2r)_+}]$. Then for any $\eta > 0$ and $K>0$ there exists $C = C(\beta , r, K) > 0$ such that if $\Vert Du_{\varepsilon }\Vert _{L^r(\varOmega )}\le K$, then

$$\begin{aligned} \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta }|Du_{\varepsilon }| \le \eta \int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta -2}|D^2 c_{\varepsilon }|^2+C~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.33)

Proof

We invoke the Hölder inequality with exponents $\frac{r}{r-1}$ and r to see that

$$\begin{aligned}\displaystyle \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta }|Du_{\varepsilon }| \le&\left( \displaystyle \int _\varOmega |\nabla c_{\varepsilon }|^{\frac{2\beta r}{r-1}}\right) ^{\frac{r-1}{r}} \left( \displaystyle \int _\varOmega |Du_{\varepsilon }|^r\right) ^{\frac{1}{r}} \\ \le&K\left( \displaystyle \int _\varOmega |\nabla c_{\varepsilon }|^{\frac{2\beta r}{r-1}}\right) ^{\frac{r-1}{r}} \\ \le&K\Vert \nabla c_{\varepsilon }\Vert _{L^{\frac{2\beta r}{r-1}}(\varOmega )}^{2\beta } ~~\text {for all}~~ t>0. \end{aligned} $$

Since $\beta \in [r-1,\frac{r-1}{(4-2r)_+}]$ ensures that $\lambda :=\frac{2\beta r}{r-1}\in [2\beta +2,4\beta +1].$ Therefore, we may apply Lemma 1.7 and (1.2.10) to see that for some $k_1 = k_1 (K,r,\beta ) > 0$ and $k_2 =k_2 (K,\beta ,r) > 0$, it has

$$ \begin{aligned} K\Vert \nabla c_{\varepsilon }\Vert _{L^{\frac{2\beta r}{r-1}}(\varOmega )}^{2\beta } \le&k_1\Vert |\nabla c_{\varepsilon }|^{\beta -1}D^2c_{\varepsilon }\Vert _{L^2(\varOmega )}^{\frac{4\beta (\lambda -3)}{(2\beta -1)\lambda }} \Vert c_{\varepsilon }\Vert _{L^\infty (\varOmega )}^{2\beta \frac{6\beta -\lambda }{(2\beta -1)\lambda }}+k_1\Vert c_{\varepsilon }\Vert _{L^\infty (\varOmega )}^{2\beta } \\ \le&k_2(\Vert |\nabla c_{\varepsilon }|^{\beta -1}D^2c_{\varepsilon }\Vert _{L^2(\varOmega )}^{\frac{4\beta (\lambda -3)}{(2\beta -1)\lambda }}+1) ~~\text {for all}~~ t>0. \end{aligned} $$

Thanks to the assumption $r >\frac{3}{2}$ and $\beta \in [r-1,\frac{r-1}{(4-2r)_+}]$, we have

$$ {\frac{4\beta (\lambda -3)}{(2\beta -1)\lambda }}={\frac{4\beta (\frac{2\beta r}{r-1}-3)}{(2\frac{2\beta r}{r-1}-1)\frac{2\beta r}{r-1}}}<2, $$

and thus arrive at (1.3.33) by means of the Young inequality.

At this position, on the basis of space–time regularity property of $n_\varepsilon$ provided by Lemmas 1.16, 1.14 can be exploited so as to derive the further regularity features of $n_\varepsilon$.

Lemma 1.20

Let $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon ,P_\varepsilon )$ be the solution of (1.2.2) as well as $\frac{10}{9}< m+\alpha \le \frac{3}{2}$. Then there exists $C>0$ such that

$$\begin{aligned} \displaystyle {\sup _{t\in (0,\infty )}\Vert n_{\varepsilon } (\cdot ,t)+\varepsilon \Vert ^{{{p_1}}}_{L^{{p_1}}(\varOmega )}+\sup _{t\in (0,\infty )} \int _t^{t+1}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{p_1+m-3} |\nabla n_{\varepsilon }|^2 \le C ,} \end{aligned}$$

(1.3.34)

where $p_1=\frac{16}{3}(m+\alpha )^2-\frac{25(m+\alpha )}{3}+4+\frac{\alpha }{3}(4(m+\alpha )-1)$.

Proof

Let $\beta _1=2m+2\alpha -1.$ Then in view of (1.3.32) and (1.2.10), we obtain that for some $C_1>0$ and all $t>0$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2\beta _1}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _1}}}_{L^{{2\beta _1}}(\varOmega )} +\frac{2(\beta _1-1)}{{\beta _1^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{\beta _1}\right| ^2 +\int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta _1-2}|D^2 c_{\varepsilon }|^2}\\ \le&\displaystyle {\Vert c_{0} \Vert _{L^\infty (\varOmega )}\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _1-1}|\nabla n_{\varepsilon }| -\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _1}+\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _1}|Du_{\varepsilon }|+C_1}. \end{aligned} \end{aligned}$$

(1.3.35)

By Lemma 1.7 and the Young inequality twice, we can conclude that for some $C_2>0$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert c_{0} \Vert _{L^\infty (\varOmega )}\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _1-1}|\nabla n_{\varepsilon }|} \\ =&\displaystyle {\Vert c_{0} \Vert _{L^\infty (\varOmega )}\int _\varOmega \left( n_{\varepsilon }^{\frac{2m+2\alpha -3}{2}}|\nabla n_{\varepsilon }|\right) \left( n_{\varepsilon }^{\frac{3-2m-2\alpha }{2}}|\nabla c_{\varepsilon }|^{\frac{3-2m-2\alpha }{2}2\beta _1}\right) |\nabla c_{\varepsilon }|^{2\beta _1-1-\frac{3-2m-2\alpha }{2}2\beta _1}} \\ \le&\frac{1}{2}\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _1} +\frac{1}{2\lambda _{0,\beta _1}}\int _\varOmega |\nabla c_{\varepsilon }|^{[2\beta _1-1-\frac{3-2m-2\alpha }{2}2\beta _1]\frac{1}{m+\alpha -1}} \\&+C_2\int _\varOmega (n_{\varepsilon }+\varepsilon )^{2m+2\alpha -3}|\nabla n_{\varepsilon }|^2 \\ =&\displaystyle {\frac{1}{2}\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _1} +\frac{1}{2\lambda _{0,\beta _1}}\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _1+2}+C_2 \int _\varOmega (n_{\varepsilon }+\varepsilon )^{2m+2\alpha -3}|\nabla n_{\varepsilon }|^2+\frac{1}{2}} \\ \le&\displaystyle {\frac{1}{2}\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _1} +\frac{1}{2}\Vert |\nabla c_\varepsilon |^{\beta _1-1}D^2c_\varepsilon \Vert _{L^2(\varOmega )}^{2}+C_2 \int _\varOmega (n_{\varepsilon }+\varepsilon )^{2m+2\alpha -3}|\nabla n_{\varepsilon }|^2+\frac{1}{2}.} \end{aligned} \end{aligned}$$

(1.3.36)

Here we have used the fact that $\frac{1}{2}+\frac{2m+2\alpha -2}{2}+\frac{3-2m-2\alpha }{2}=1.$ Inserting (1.3.36) into (1.3.35), we then have

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2\beta _1}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _1}}}_{L^{{2\beta _1}}(\varOmega )} +\frac{2(\beta _1-1)}{{\beta _1^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{\beta _1}\right| ^2} \\&+\frac{1}{2}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta _1-2}|D^2 c_{\varepsilon }|^2 +\frac{1}{2}\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _1} \\ \le&\displaystyle {C_2\int _\varOmega (n_{\varepsilon }+\varepsilon )^{2m+2\alpha -3}|\nabla n_{\varepsilon }|^2 +\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _1}|Du_{\varepsilon }|+\frac{1}{2}.} \end{aligned} \end{aligned}$$

(1.3.37)

In addition, it is observed that

$$ \frac{3(m+\alpha )}{3-(m+\alpha )}-1<\beta _1<\frac{\frac{3(m+\alpha )}{3-(m+\alpha )}-1}{(4-2\frac{3(m+\alpha )}{3-(m+\alpha )})_+} $$

can be warranted by $m+\alpha \in (\frac{10}{9},\frac{3}{2}]$, and thereby by Lemmas 1.16 and 1.17, there exists a constant $C_3>0$ such that

$$ \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _1}|Du_{\varepsilon }| \le \frac{1}{4}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta _1-2}|D^2 c_{\varepsilon }|^2+C_3~~\text {for all}~~ t>0. $$

Substituting it into (1.3.37), one immediately obtains that for some $C_4>0$

$$ \begin{aligned}&\displaystyle {\frac{1}{{2\beta _1}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _1}}}_{L^{{2\beta _1}}(\varOmega )} +\frac{2(\beta _1-1)}{{\beta _1^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{\beta _1}\right| ^2}\\&\displaystyle {+\frac{1}{4}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta _1-2}|D^2 c_{\varepsilon }|^2+\frac{1}{2} \int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _1}} \\ \le&\displaystyle {C_2\int _\varOmega (n_{\varepsilon }+\varepsilon )^{2m+2\alpha -3}|\nabla n_{\varepsilon }|^2+C_4,} \end{aligned} $$

which along with (1.3.29) leads to

$$\begin{aligned} \begin{aligned}&\displaystyle {\sup _{t\in (0,\infty )}\Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _1}}}_{L^{{2\beta _1}}(\varOmega )} +\sup _{t\in (0,\infty )}\int _t^{t+1} \int _{\varOmega } (|\nabla c_{\varepsilon }|^{2\beta _1-2}|D^2 c_{\varepsilon }|^2+n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _1})\le C_5 } \end{aligned} \end{aligned}$$

(1.3.38)

for some $C_5> 0$.

Moreover, denoting $p_0=m+2\alpha$ and taking $p:=p_1=\frac{16}{3}(m+\alpha )^2 -\frac{25}{3}(m+\alpha )+4+\frac{1}{3}\alpha [4(m+\alpha )-1]$ in (1.3.26), and applying the Young inequality, we conclude that for any $\delta >0,$ there exists constant $C(\delta )>0$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{p_{1}}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p_1}}}_{L^{{p_1}}(\varOmega )}+ \frac{{m}(p_{1}-1)}{2}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p_1-3} |\nabla n_{\varepsilon }|^2}\\ \le&\displaystyle {C_1\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p_{1}+1-m-2\alpha }|\nabla c_{\varepsilon }|^2 }\\ \le&\displaystyle {\delta \int _\varOmega (n_{\varepsilon }+\varepsilon )^{[{p_{1}+1-m-2\alpha }-\frac{1}{\beta _{1}}]\frac{\beta _{1}}{\beta _{1}-1}}+C(\delta )\int _\varOmega (n_{\varepsilon }+\varepsilon )|\nabla c_{\varepsilon }|^{2\beta _{1}} }\\ \le&\displaystyle {\delta \int _\varOmega (n_{\varepsilon }+\varepsilon )^{\frac{5m}{3}+p_1-1+\frac{4\alpha }{3}}+C(\delta )\int _\varOmega (n_{\varepsilon }+\varepsilon )|\nabla c_{\varepsilon }|^{2\beta _1}, } \end{aligned} \end{aligned}$$

(1.3.39)

thanks to $(p_1+1-m-2\alpha -\frac{1}{\beta _1})\frac{\beta _1}{\beta _1-1}= \frac{5m}{3}+p_1-1+\frac{4\alpha }{3}=m+p_1-1+\frac{2}{3}p_0.$

Further, by the Gagliardo–Nirenberg interpolation inequality, we infer from (1.3.29) that

$$ \begin{aligned}&\displaystyle {\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+p_1-1+\frac{2}{3}p_0}} \\ =&\displaystyle {\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p_1-1}{2}}\Vert ^{\frac{2(m+p_{1}-1+\frac{2}{3}p_{0})}{m+p_1-1}}_{L^{\frac{2(m+p_{1}-1 +\frac{2}{3}p_{0})}{m+p_{1}-1}} (\varOmega )}} \\ \le&C_6\left( \Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{p_{1}+m-1}{2}}\Vert _{L^2(\varOmega )}^2 \Vert (n_{\varepsilon }+\varepsilon )^{\frac{p_{1}+m-1}{2}}\Vert _{L^{\frac{2p_{0}}{m+p_{1}-1}}(\varOmega )}^{\frac{2(m+p_{2}-1+\frac{2}{3}p_{0})}{m+p_{1}-1}-2}\right. \\&+\left. \Vert (n_{\varepsilon } +\varepsilon )^{\frac{p_{1}+m-1}{2}}\Vert _{L^{\frac{2p_{0}}{m+p_{1}-1}}(\varOmega )}^{\frac{2(m+p_{1}-1+\frac{2}{3}p_{0})}{m+p_{1}-1}}\right) \\ \le&\displaystyle {C_7(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{p_{1}+m-1}{2}}\Vert _{L^2(\varOmega )}^2+1)} \end{aligned} $$

with constants $C_6>0$ and $C_7>0$. Inserting the above inequality into (1.3.29) and picking $\delta >0$ appropriately small, one concludes that there exists a positive constant $C_8$ such that

$$ \begin{aligned}&\displaystyle {\frac{1}{{p_{1}}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p_1}}}_{L^{{p_1}}(\varOmega )}+ \displaystyle \frac{{m}(p_1-1)}{4}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p_1-3} |\nabla n_{\varepsilon }|^2} \displaystyle {+\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p_1}}\\ \le&\displaystyle {C_8\int _\varOmega n_{\varepsilon }|\nabla c_{\varepsilon }|^{2\beta _{1}}+ \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _{1}+2}+C_8.} \end{aligned} $$

Now by (1.3.38) and (1.2.8), one can get

$$ \displaystyle {\sup _{t\in (0,\infty )}\Vert n_{\varepsilon }(\cdot ,t)+\varepsilon \Vert ^{{{p_1}}}_{L^{{p_1}}(\varOmega )}+\sup _{t\in (0,\infty )} \int _t^{t+1}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{p_1+m-3} |\nabla n_{\varepsilon }|^2 \le C_9} $$

for some positive constant $C_9$.

Lemma 1.21

Let $\frac{10}{9}<m+\alpha \le \frac{3}{2}$ and $(n_\varepsilon ,c_\varepsilon ,u_\varepsilon )$ be the solution of (1.2.2). Then for any $\beta >1$ and $\eta >0$ there exists a constant $C = C(\beta , \eta ) > 0$ such that

$$ \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta } +\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta }|Du_{\varepsilon }| \le \eta \int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta -2}|D^2 c_{\varepsilon }|^2+C~~\text {for all}~~ t>0. $$

Proof

It is observed that $m+\alpha >\frac{10}{9}$ ensures $p_1=\frac{16}{3}(m+\alpha )^2-\frac{25}{3}(m+\alpha )+4+\frac{1}{3}\alpha [4(m+\alpha )-1]>\frac{322}{243}$, and thus from Lemma 1.20, we have

$$ \displaystyle {\sup _{t\in (0,\infty )}\Vert n_{\varepsilon } (\cdot ,t)\Vert _{L^{{\frac{322}{243}}}(\varOmega )}\le C_1. } $$

By Lemma 1.3, for any $2<r<\frac{966}{407}$, $\Vert D u_{\varepsilon }(\cdot , t)\Vert _{L^r(\varOmega )}\le C_2$ for some positive constant $C_2$. Moreover, by Lemma 1.19, one can conclude that for any $\beta >1$

$$\begin{aligned} \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta }|Du_{\varepsilon }|\le \frac{\eta }{2} \int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta -2}|D^2 c_{\varepsilon }|^2+C_3~~\text {for all}~~ t>0 \end{aligned}$$

(1.3.40)

with some positive constant $C_3$. On the other hand, in view of Lemma 1.7, it follows from the Young inequality that there is $C_4>0$ satisfying

$$ \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta }\le \frac{\eta }{2} \int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta -2}|D^2 c_{\varepsilon }|^2+C_4~~\text {for all}~~ t>0, $$

which together with (1.3.40) leads to the desired inequality.

The regularity of $\nabla c_\varepsilon$ from Lemma 1.21 can be readily developed to the following basis for the iterative reason, which can elevate $L^{p_1}(\varOmega )$ of $n_\varepsilon$ from Lemma 1.20 to the $L^p(\varOmega )$-boundedness of $n_\varepsilon$ with some $p>\frac{3}{2}$. To this end, we consider the properties of the iteration sequence $\{p_n\}_{n\ge 1}$ stated in the following.

Lemma 1.22

Let $p_1= \frac{16}{3}(m+\alpha )^2-\frac{25}{3}(m+\alpha )+\frac{1}{3}\alpha (4(m+\alpha )-1)+4,$ $\frac{10}{9}< m\le \frac{3}{2}$ and $0\le \alpha \le \frac{7}{18}$. Assume that for any $n=1,2,\cdots$,

$$ p_{n+1}=\frac{2}{3}p_{n}^2+\frac{2}{3}(4m-5+3\alpha )p_n+(2m+2\alpha -3)(m-1)+1, $$

then $p_1$ is monotonically non-decreasing functions with respect to m as well as $p_n$ is monotonically non-decreasing functions with respect to n, inter alia $\displaystyle \lim _{n\rightarrow \infty }p_n=+\infty$.

Proof

A direct calculation shows that

$$ \begin{aligned} p_{n+1}=&\frac{2}{3}p_{n}^2+\frac{2}{3}(4m-5+3\alpha )p_n+(2m+2\alpha -3)(m-1)+1\\ =&\frac{2}{3}p_n[p_n+(4m-5)+3\alpha ]+(2m+2\alpha -3)(m-1)+1.\\ \end{aligned} $$

Due to $0<\alpha \le \frac{7}{18}$ and $\frac{10}{9}<m+\alpha \le \frac{3}{2}$, the mathematical induction implies that for any $n\in \mathbb {N}^*$, $p_n\ge m+\alpha .$ In addition, it is observed that

$$ \begin{aligned} p_{n+1}-p_{n}=&\displaystyle {\frac{2}{3}(p_n^2-p_{n-1}^2)+\frac{2}{3}(4m-5)(p_n-p_{n-1})} \\ =&\displaystyle {\frac{2}{3}(p_n-p_{n-1})(p_n+p_{n-1}+4m-5+3\alpha )} . \end{aligned} $$

Hence in light of $p_2>p_1$ and $m+\alpha >\frac{10}{9}$, one can see that $p_3>p_2$, and thereby $p_{n+1}>p_{n}$ by the induction.

By the contradiction argument, one can show that $\displaystyle \lim _{n\rightarrow \infty }p_n=+\infty$. In fact, supposed that $\{p_n\}_{n\ge 1}$ is bounded, then $\lim _{n\rightarrow \infty }p_n=p_*$ with some positive constant $p_*<\infty$, which implies that $p_{*}=\frac{2}{3}p_{*}^2+\frac{2}{3}(4m-5+3\alpha )p_*+(2m+2\alpha -3)(m-1)+1,$ that is

$$\begin{aligned} \frac{2}{3}p_*^2+(\frac{8}{3}m-\frac{13}{3}+2\alpha )p_*+(2m+2\alpha -3)(m-1)+1=0. \end{aligned}$$

(1.3.41)

By Weda’s Theorem, we have

$$\begin{aligned} \begin{aligned} 0\le \triangle =&(\frac{8}{3}m-\frac{13}{3}+2\alpha )^2-4\times \frac{2}{3}\times [(2m+2\alpha -3)(m-1)+1]\\ =&\displaystyle \frac{16\rho ^2-8(11-2\alpha )\rho +4\alpha ^2-20\alpha +73}{9}\\ :=&\displaystyle \frac{H(\rho ,\alpha )}{9} \end{aligned} \end{aligned}$$

(1.3.42)

with $\rho =m+\alpha$. Note that for any $0\le \alpha \le \frac{7}{18}$ and $\frac{10}{9}<m+\alpha \le \frac{3}{2}$, $\frac{\partial H(\rho ,\alpha )}{\partial \rho }=32\rho -8(11-2\alpha )<0$. So for $0\le \alpha \le \frac{7}{18}$ and $\frac{10}{9}<m+\alpha \le \frac{3}{2}$,

$$ H(\rho ,\alpha )\le H(\frac{10}{9},\alpha ) =16(\frac{10}{9})^2-8(11-2\alpha )\frac{10}{9}+4\alpha ^2-20\alpha +73<0, $$

which contradicts with (1.3.42).

Lemma 1.23

Let $\frac{10}{9}<m+\alpha \le \frac{3}{2}$ as well as $0\le \alpha \le \frac{7}{18}$. If

$$\begin{aligned} \displaystyle {\sup _{t\in (0,\infty )}\Vert n_{\varepsilon }(\cdot ,t)+\varepsilon \Vert ^{{{p_{n}}}}_{L^{{p_{n}}}(\varOmega )}+\sup _{t\in (0,\infty )} \int _t^{t+1}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{p_{n}+m-3} |\nabla n_{\varepsilon }|^2 \le K} \end{aligned}$$

(1.3.43)

with ${p_{n}+m-3}< 0$ for some $K>0$, then there exists $C=C(K)>0$ independent of $\varepsilon$, such that

$$ \displaystyle {\sup _{t\in (0,\infty )}\Vert n_{\varepsilon }(\cdot ,t)+\varepsilon \Vert ^{{{p_{n+1}}}}_{L^{{p_{n+1}}}(\varOmega )}+\sup _{t\in (0,\infty )} \int _t^{t+1}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{p_{n+1}+m-3} |\nabla n_{\varepsilon }|^2 \le C,} $$

where $p_{n+1}=\frac{2}{3}p_{n}^2+2(\frac{4}{3}m-\frac{5}{3}+\alpha )p_n+(2m+2\alpha -3)(m-1)+1$ for any $n=1,2,3,\cdots$, and $p_1$ is taken from Lemma 1.20.

Proof

Let $\beta _n=p_n+m-1~~~\text {for any}~~n=1,2,3,\cdots .$ Recalling (1.3.32), there is $C_1>0$ such that for all $t>0$,

$$\begin{aligned} \begin{aligned}&\displaystyle { \frac{1}{2\beta _n} \frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{2\beta _n}_{L^{2\beta _n}(\varOmega )} } + \displaystyle {\frac{2(\beta _n-1)}{{\beta _n^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{\beta _n}\right| ^2 +\int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta _n-2}|D^2 c_{\varepsilon }|^2}\\&\le \displaystyle {\Vert c_0\Vert _{L^\infty (\varOmega )}\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _n-1}|\nabla n_{\varepsilon }|} -\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _n}+\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _n}|Du_{\varepsilon }|+C_1. \end{aligned} \end{aligned}$$

(1.3.44)

As done in (1.3.36), we can conclude that there exists a positive constant $C_2$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert c_{0} \Vert _{L^\infty (\varOmega )}\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _n-1}|\nabla n_{\varepsilon }|} \\ \le&\displaystyle {\frac{1}{2}\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _n}+\frac{1}{2}\Vert |\nabla c_\varepsilon |^{\beta _n-1}D^2c_\varepsilon \Vert _{L^2(\varOmega )}^{2}} \\&+C_2 \int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+p_n-3}|\nabla n_{\varepsilon }|^2+\frac{1}{2}, \end{aligned} \end{aligned}$$

(1.3.45)

which along with (1.3.44) implies that

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2\beta _n}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _n}}}_{L^{{2\beta _n}}(\varOmega )} +\frac{2(\beta _n-1)}{{\beta _n^2}}\int _{\varOmega }\left| \nabla |\nabla c_{\varepsilon }|^{\beta _n}\right| ^2} \\&+\frac{1}{2}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta _n-2}|D^2 c_{\varepsilon }|^2 +\frac{1}{2}\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _n} \\ \le&\displaystyle {C_2\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+p_n-3}|\nabla n_{\varepsilon }|^2 +\int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _n}|Du_{\varepsilon }|+\frac{1}{2}} \end{aligned} \end{aligned}$$

(1.3.46)

and thereby together with Lemma 1.21 leads to

$$ \begin{aligned}&\displaystyle {\frac{1}{{2\beta _n}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _n}}}_{L^{{2\beta _n}}(\varOmega )} +\Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _n}}}_{L^{{2\beta _n}}(\varOmega )}+\frac{1}{8}\int _{\varOmega } |\nabla c_{\varepsilon }|^{2\beta _n-2}|D^2 c_{\varepsilon }|^2 +\frac{1}{2}\int _{\varOmega } n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _n}} \\ \le&\displaystyle {C_2 \int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+p_n-3}|\nabla n_{\varepsilon }|^2+C_4} \end{aligned} $$

with some constant $C_4>0$. By (1.3.43), there exists some positive constant $C_5$ such that

$$\begin{aligned} \displaystyle {\sup _{t\in (0,\infty )}\Vert \nabla c_{\varepsilon }\Vert ^{{{2\beta _n}}}_{L^{{2\beta _n}}(\varOmega )} +\sup _{t\in (0,\infty )}\int _t^{t+1}\int _{\varOmega } [|\nabla c_{\varepsilon }|^{2\beta _n-2}|D^2 c_{\varepsilon }|^2+n_\varepsilon |\nabla c_{\varepsilon }|^{2\beta _n}]\le C_5 }. \end{aligned}$$

(1.3.47)

Furthermore, in view of $p_n>m+\alpha$, taking $p_{n+1}=\frac{2}{3}p_{n}^2+2(\frac{4}{3}m-\frac{5}{3}+\alpha )p_n+(2m+2\alpha -3)(m-1)+1$ in (1.3.26) and by the Young inequality, it follows that for any $\eta >0,$ there is $C(\eta )>0$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{p_{n+1}}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p_{n+1}}}}_{L^{{p_{n+1}}}(\varOmega )}+ \frac{{m}(p_{n+1}-1)}{2}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p_{n+1}-3}(1+n_{\varepsilon })^{-2\alpha } |\nabla n_{\varepsilon }|^2}\\ \le&\displaystyle {\frac{(p_{n+1}-1)C_S^2}{2{m}}\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p_{n+1}+1-m-2\alpha }|\nabla c_{\varepsilon }|^2 }\\ =&\displaystyle {\frac{(p_{n+1}-1)C_S^2}{2{m}}\int _\varOmega (n_{\varepsilon }+\varepsilon )^{\frac{1}{\beta _{n}}}|\nabla c_{\varepsilon }|^2 (n_{\varepsilon }+\varepsilon )^{p_{n+1}+1-m-2\alpha -\frac{1}{\beta _{n}}} }\\ \le&\displaystyle {C(\eta )\int _\varOmega (n_{\varepsilon }+\varepsilon )|\nabla c_{\varepsilon }|^{2\beta _{n}} + \eta \int _\varOmega (n_{\varepsilon }+\varepsilon )^{[p_{n+1}+1-m-2\alpha -\frac{1}{\beta _{n}}]\frac{\beta _{n}}{\beta _{n}-1}} }. \end{aligned} \end{aligned}$$

(1.3.48)

Thanks to $p_{n+1}=\frac{2}{3}p_{n}^2+2(\frac{4}{3}m-\frac{5}{3}+\alpha )p_n+(2m+2\alpha -3)(m-1)+1$, we can see that

$$ \begin{aligned} (p_{n+1}+1-m-2\alpha -\frac{1}{\beta _{n}})\frac{\beta _{n}}{\beta _{n}-1} =&(p_{n+1}+1-m-2\alpha -\frac{1}{p_n+m-1})\frac{p_n+m-1}{p_n+m-2}\\ =&{m+p_{n+1}-1+\frac{2}{3}p_{n}} \end{aligned} $$

and thereby

$$ \begin{aligned}&\displaystyle {\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+p_{n+1}-1+\frac{2}{3}p_{n}}} \\ =&\displaystyle {\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p_{n+1}-1}{2}}\Vert ^{\frac{2(m+p_{n+1}-1 +\frac{2}{3}p_{n})}{m+p_{n+1}-1}}_{L^{\frac{2(m+p_{n+1}-1+\frac{2}{3}p_{n})}{m+p_{n+1}-1}}(\varOmega )}} \\ \le&\displaystyle {C_6\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{p_{n+1}+m-1}{2}}\Vert _{L^2(\varOmega )}^2 \Vert (n_{\varepsilon }+\varepsilon )^{\frac{p_{n+1}+m-1}{2}} \Vert _{L^{\frac{2p_{n}}{m+p_{n+1}-1}}(\varOmega )}^{\frac{2(m+p_{n+1}-1+\frac{2}{3}p_{n})}{m+p_{n+1}-1}-2}} \\&\displaystyle {+C_6\Vert (n_{\varepsilon }+\varepsilon )^{\frac{p_{n+1}+m-1}{2}}\Vert _{L^{\frac{2p_{n}}{m+p_{n+1}-1}} (\varOmega )}^{\frac{2(m+p_{n}-1+\frac{2}{3}p_{n})}{m+p_{n}-1}}}\\ \le&\displaystyle {C_7(\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{p_{n+1}+m-1}{2}}\Vert _{L^2(\varOmega )}^2+1).} \end{aligned} $$

Substituting the above inequality into (1.3.48) and taking $\eta >0$ appropriately small, one may derive that there is $C_8>0$ such that for any $\varepsilon \in (0,1),$

$$\begin{aligned}&\displaystyle {\frac{1}{{p_{n+1}}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{p_{n+1}}}}_{L^{{p_{n+1}}}(\varOmega )}+ \displaystyle \frac{{m}(p_{n+1}-1)}{4}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p_{n+1}-3} |\nabla n_{\varepsilon }|^2} \nonumber \\&+\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+p_{n+1}-1+\frac{2}{3}p_{n}} \nonumber \\ \le&\displaystyle {C(\eta )\int _\varOmega n_{\varepsilon }|\nabla c_{\varepsilon }|^{2\beta _{n}}+ \int _\varOmega |\nabla c_{\varepsilon }|^{2\beta _{n}}+C_8.} \nonumber \end{aligned}$$

This together with (1.3.47) implies that for some positive constant $C_9$,

$$ \begin{aligned}&\displaystyle {\sup _{t\in (0,\infty )}\Vert n_{\varepsilon }(\cdot ,t)+\varepsilon \Vert ^{{{p_{n+1}}}}_{L^{{p_{n+1}}}(\varOmega )}+\sup _{t\in (0,\infty )} \int _t^{t+1}\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{p_{n+1}+m-3} |\nabla n_{\varepsilon }|^2 \le C_{9} } \end{aligned} $$

and thus completes the proof of Lemma 1.23.

Combining Lemma 1.3.34 with Lemma 1.23, we immediately have

Lemma 1.24

Let $0\le \alpha \le \frac{7}{18}$ and $\frac{10}{9}<m+\alpha \le \frac{3}{2}$. Then there exist constants $p^*>\frac{3}{2}$ and $C=C(p^*)>0$ such that

$$ \int _{\varOmega }n^{p^*}_\varepsilon (x,t)dx\le C $$

for all $t>0$.

By the similar strategy as above, one can also derive the boundedness of $\int _{\varOmega }n^{p}_\varepsilon$ with some $p>\frac{3}{2}$ in the case $m+\alpha >2$.

Lemma 1.25

Let $m+\alpha >2$. There exists $C>0$ independent of $\varepsilon$ such that the solution of (1.2.2) satisfies

$$\begin{aligned} \displaystyle {\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+2\alpha -1}\le C~~~v{for all}~~ t>0} \end{aligned}$$

(1.3.49)

as well as

$$\begin{aligned} \displaystyle {\int _{t}^{t+1}\int _{\varOmega } \left[ (n_{\varepsilon }+\varepsilon )^{2(m+\alpha -\frac{2}{3})} + (n_{\varepsilon }+\varepsilon )^{2m+2\alpha -4} |\nabla {n_{\varepsilon }}|^2\right] \le C.} \end{aligned}$$

(1.3.50)

Proof

Taking ${c_{\varepsilon }}$ as the test function for the second equation of (1.2.2) and using $\nabla \cdot u_\varepsilon =0$, it yields that

$$ \displaystyle {\frac{1}{{2}}\frac{d}{dt}\Vert {c_{\varepsilon }}\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega } |\nabla c_{\varepsilon }|^2 =-\int _{\varOmega } n_{\varepsilon }c^2_{\varepsilon },} $$

which together with $n_{\varepsilon }\ge 0$ and $c_{\varepsilon }\ge 0$ implies that for some positive constant $C_1$,

$$ \displaystyle {\int _{\varOmega } c_{\varepsilon }^2+\int _{t}^{t+1}\int _{\varOmega } |\nabla {c_{\varepsilon }}|^2\le C_1~~\text {for all}~~ t>0}. $$

Now, choosing $p=m+2\alpha -1$ in (1.3.26), we get

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{m+2\alpha -1}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{m+2\alpha -1}}}_{L^{{m+2\alpha -1}}(\varOmega )}+ \frac{{m}(m+2\alpha -2)}{2}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -4} |\nabla n_{\varepsilon }|^2}\\ \le&\displaystyle {\frac{(m+2\alpha -2)C_S^2}{2m}\int _\varOmega (n_{\varepsilon }+\varepsilon )^{2\alpha }(1+n_{\varepsilon })^{-2\alpha }|\nabla c_{\varepsilon }|^2}\\ \le&\displaystyle {\frac{(m+2\alpha -2)C_S^2}{2m}\int _\varOmega |\nabla c_{\varepsilon }|^2~~~\text {for all}~~ t>0.} \end{aligned} \end{aligned}$$

(1.3.51)

Furthermore, applying the Gagliardo–Nirenberg inequality, we obtain that there are $C_i>0, (i=1,2,3),$ such that

$$ \begin{aligned}&\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+2\alpha -1} \\ =&\displaystyle {\Vert (n_{\varepsilon }+\varepsilon )^{m+\alpha -1}\Vert ^{\frac{m+2\alpha -1}{m+\alpha -1}}_{L^\frac{m+2\alpha -1}{m+\alpha -1}(\varOmega )}}\\ \le&\displaystyle {C_1\Vert \nabla (n_{\varepsilon }+\varepsilon )^{m+\alpha -1}\Vert ^{2\frac{3(m+2\alpha -2)}{6m+6\alpha -7}}_{L^2(\varOmega )} \Vert (n_{\varepsilon }+\varepsilon )^{m+\alpha -1}\Vert ^{{\frac{m+2\alpha -1}{m+\alpha -1}}-2\frac{3(m+2\alpha -2)}{6m+6\alpha -7}}_{L^\frac{1}{m+\alpha -1}(\varOmega )}}\\&\displaystyle {+C_1 \Vert (n_{\varepsilon }+\varepsilon )^{m+\alpha -1}\Vert ^{{\frac{m+2\alpha -1}{m+\alpha -1}}}_{L^\frac{1}{m+\alpha -1}(\varOmega )}}\\ \le&\displaystyle {C_2( \Vert \nabla (n_{\varepsilon }+\varepsilon )^{m+\alpha -1}\Vert ^{\frac{6(m+2\alpha -2)}{6m+6\alpha -7}}_{L^2(\varOmega )}+1)}\\ \le&\displaystyle {\frac{{m}(m+2\alpha -2)}{8}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -4} |\nabla n_{\varepsilon }|^2+C_3} \end{aligned} $$

thanks to $\frac{6(m+2\alpha -2)}{6m+6\alpha -7}<2$, and

$$\begin{aligned} \begin{aligned} \displaystyle \int _\varOmega (n_{\varepsilon }+\varepsilon )^{2(m+\alpha -\frac{2}{3})}=&\Vert (n_{\varepsilon }+\varepsilon )^{m+\alpha -1}\Vert ^{\frac{{2(m+\alpha -\frac{2}{3})}}{m+\alpha -1}}_{L^ \frac{2(m+\alpha -\frac{2}{3})}{m+\alpha -1}(\varOmega )} \\ \le&\displaystyle C_3( \Vert \nabla (n_{\varepsilon }+\varepsilon )^{m+\alpha -1}\Vert ^{2}_{L^2(\varOmega )}+1). \end{aligned} \end{aligned}$$

Inserting above two inequalities into (1.3.51), we derive

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{m+2\alpha -1}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{m+2\alpha -1}}}_{L^{{m+2\alpha -1}}(\varOmega )} +\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+2\alpha -1}}\\&\displaystyle {+\frac{{m}(m+2\alpha -2)}{2}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -4} |\nabla n_{\varepsilon }|^2 +(n_{\varepsilon }+\varepsilon )^{2(m+\alpha -\frac{2}{3})}}\\ \le&\displaystyle {\frac{(m+2\alpha -2)C_S^2}{2m}\int _\varOmega |\nabla c_{\varepsilon }|^2+C_4~~~\text {for all}~~ t>0} \end{aligned} \end{aligned}$$

(1.3.52)

with some positive constant $C_4$. Now, we define $y_\varepsilon (t):=\Vert n_{\varepsilon }(\cdot ,t)+\varepsilon \Vert ^{{{m+2\alpha -1}}}_{L^{{m+2\alpha -1}}(\varOmega )}$ and

$$ h_\varepsilon (t):=\frac{{m}(m+2\alpha -2)}{2}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -4} |\nabla n_{\varepsilon }|^2+(n_{\varepsilon }+\varepsilon )^{2(m+\alpha -\frac{2}{3})}~~~\text {for all}~~ t>0. $$

As an application of Lemma 1.4, this together with (1.3.51) readily yields (1.3.49) and (1.3.50).

With the space–time regularity property of $n_{\varepsilon }$ in (1.3.50), we can improve the regularity of $\nabla u_{\varepsilon }$ beyond (1.3.14) through following lemma.

Lemma 1.26

Let $m+\alpha >2$. There exists constant $C>0$ such that for all $t>0$,

$$\begin{aligned} \int _{\varOmega }|\nabla u_{\varepsilon }(\cdot ,t)|^2+ \int _t^{t+1}\int _{\varOmega }|\varDelta u_{\varepsilon }|^2\le C. \end{aligned}$$

(1.3.53)

Proof

Multiplying the projected Stokes equation $u_{\varepsilon t} +Au_{\varepsilon } = \mathscr {P}[n_{\varepsilon }\nabla \phi ]$ by $Au_{\varepsilon }$, we derive

$$\begin{aligned} \begin{aligned} \displaystyle {\frac{1}{{2}}\frac{d}{dt}\Vert A^{\frac{1}{2}}u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}+ \int _{\varOmega }|Au_{\varepsilon }|^2 }=&\displaystyle { \int _{\varOmega }\mathscr {P}(n_{\varepsilon }\nabla \phi ) Au_{\varepsilon }}\\ \le&\displaystyle { \frac{1}{2}\int _{\varOmega }|Au_{\varepsilon }|^2+ \frac{1}{2}\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }n_{\varepsilon }^2~~\text {for all}~~ t>0.} \end{aligned} \end{aligned}$$

(1.3.54)

Recalling that $\Vert A^{\frac{1}{2}}u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )} = \Vert \nabla u_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}$ (see p. 133 of Sohr 2001), and with some $C _1> 0$, we have

$$ \int _{\varOmega }{|\nabla u_{\varepsilon }(\cdot ,t)|^2}\le C_1 \int _{\varOmega }|Au_{\varepsilon }|^2~~\text {for all}~~ t>0. $$

Thanks to the fact that $\Vert \cdot \Vert _{W^{2,2}(\varOmega )}$ and $\Vert A(\cdot )\Vert _{L^{2}(\varOmega )}$ are equivalent on D(A) (see p. 129 of Sohr 2001), we see that for

$$y (t) :=\int _{\varOmega }{|\nabla u_{\varepsilon }(\cdot ,t)|^2}, t >0 $$

and

$$ h(t) := \frac{1}{2}\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _{\varOmega }n_{\varepsilon }^2,t >0. $$

Equation (1.3.54) implies the inequality

$$ y'(t)+\frac{1}{2C_1}y(t)+\frac{1}{2}\int _{\varOmega }|Au_{\varepsilon }|^2\le h(t)~~\text {for all}~~ t>0. $$

As an application of Lemma 1.4, this yields (1.3.53) thanks to (1.3.50).

At this position, we can achieve the regularity of $c_{\varepsilon }$ in the case $m+\alpha >2$ just as that in Lemma 1.13.

Lemma 1.27

Let $m+\alpha >2$. There exists $C>0$ independent of $\varepsilon$ such that

$$\begin{aligned} \int _{\varOmega } |\nabla c_{\varepsilon }(\cdot ,t)|^2 +\int _t^{t+1} \displaystyle \int _{\varOmega }|\varDelta c_{\varepsilon }|^2\le C ~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.55)

Proof

Similar to the proof (1.3.42), we can conclude that

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}} \\ \le&\displaystyle {-\frac{1}{2}\int _{\varOmega } |\varDelta c_{\varepsilon }|^2 + \Vert c_{0}\Vert _{L^\infty (\varOmega )}\int _\varOmega n_{\varepsilon }^2 -\int _{\varOmega }\nabla c_{\varepsilon } (\nabla u_{\varepsilon } \cdot \nabla c_{\varepsilon })}\\ \le&\displaystyle {-\frac{1}{2}\int _{\varOmega } |\varDelta c_{\varepsilon }|^2 + \Vert c_{0}\Vert _{L^\infty (\varOmega )}\int _\varOmega n_{\varepsilon }^2 +\Vert \nabla u_{\varepsilon } \Vert _{L^{2}(\varOmega )}\Vert \nabla c_{\varepsilon } \Vert _{L^{4}(\varOmega )}^2~~\text {for all}~~ t>0.} \end{aligned} \end{aligned}$$

(1.3.56)

Recalling (1.2.10), the Gagliardo–Nirenberg inequality entails that there exist $C_1>0$ and $C_2> 0$ such that

$$\begin{aligned} \begin{aligned} \Vert \nabla c_{\varepsilon } \Vert ^2_{L^4(\varOmega )} \le&\displaystyle C_1\Vert \varDelta c_{\varepsilon } \Vert _{L^2(\varOmega )} \Vert c_{\varepsilon } \Vert ^2_{L^{\infty }(\varOmega )} +C_1\Vert c_{\varepsilon }\Vert ^2_{L^{\infty }(\varOmega )}\\ \le&\displaystyle {C_{2}\Vert \varDelta c_{\varepsilon } \Vert _{L^{2}(\varOmega )}+C_{2} ~~\text {for all}~~ t>0.} \end{aligned} \end{aligned}$$

(1.3.57)

Substituting (1.3.57) into (1.3.56) and by the Young inequality, we obtain that for some positive constant $C_3$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )}}\\ \le&\displaystyle {-\frac{1}{2}\int _{\varOmega } |\varDelta c_{\varepsilon }|^2+ \Vert c_{0}\Vert _{L^\infty (\varOmega )} \int _\varOmega n_{\varepsilon }^2+\Vert \nabla u_{\varepsilon } \Vert _{L^{2}(\varOmega )}[C_{2}\Vert \varDelta c_{\varepsilon } \Vert ^{\frac{1}{2}}_{L^{2}(\varOmega )} +C_{2}]^2}\\ \le&\displaystyle {-\frac{1}{4}\int _{\varOmega } |\varDelta c_{\varepsilon }|^2+ \Vert c_{0}\Vert _{L^\infty (\varOmega )} \int _\varOmega n_{\varepsilon }^2+C_3\Vert \nabla u_{\varepsilon } \Vert _{L^{2}(\varOmega )}^2 +C_3~~\text {for all}~~ t>0,} \end{aligned} \end{aligned}$$

(1.3.58)

which together with (1.3.57) implies that for some positive constants $C_4, C_5$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{2}}\frac{d}{dt} \Vert \nabla c_{\varepsilon }\Vert ^{{{2}}}_{L^{{2}}(\varOmega )} +\frac{1}{8}\Vert \varDelta c_{\varepsilon } \Vert _{L^{2}(\varOmega )}^{2}}+C_{4}\Vert \nabla c_{\varepsilon } \Vert _{L^{2}(\varOmega )}^2\\ \le&\displaystyle { \Vert c_{0}\Vert _{L^\infty (\varOmega )} \int _\varOmega n_{\varepsilon }^2+C_3\Vert \nabla u_{\varepsilon } \Vert _{L^{2}(\varOmega )}^2 +C_5~~\text {for all}~~ t>0.} \end{aligned} \end{aligned}$$

(1.3.59)

Now, we define $g_\varepsilon (t):=\Vert \nabla c_{\varepsilon }(\cdot ,t) \Vert _{L^{2}(\varOmega )}^2$ and

$$ h_\varepsilon (t):=\Vert c_{0}\Vert _{L^\infty (\varOmega )}\int _\varOmega n_{\varepsilon }^2(\cdot ,t)+C_3\Vert \nabla u_{\varepsilon } (\cdot ,t)\Vert _{L^{2}(\varOmega )}^2 +C_5. $$

As an application of Lemma 1.4, this in conjunction with (1.3.53) entails (1.3.55).

Proceeding as the proof of Lemma 1.16, we can arrive at

Lemma 1.28

Let $m+\alpha >2$. Then there exists $C>0$ independent of $\varepsilon$ such that the solution of (1.2.2) satisfies

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+2\alpha -\frac{1}{2}}\le C~~~\text {for all}~~ t>0.}\\ \end{aligned} \end{aligned}$$

(1.3.60)

Proof

Choosing $p = m+2\alpha -\frac{1}{2}$ in (1.3.26), we obtain that for some $C_1>0$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{1}{{m+2\alpha -\frac{1}{2}}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{m+2\alpha -\frac{1}{2}}}}_{L^{{m+2\alpha -\frac{1}{2}}}(\varOmega )} +\frac{{m}(m+2\alpha -\frac{3}{2})}{2}\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -\frac{1}{2}-3} |\nabla n_{\varepsilon }|^2}\\ \le&\displaystyle {C_1\int _\varOmega (n_{\varepsilon }+\varepsilon )^{\frac{1}{2}+2\alpha }(1+n_{\varepsilon })^{-2\alpha }|\nabla c_{\varepsilon }|^2 }\\ \le&\displaystyle {C_1\int _\varOmega (n_{\varepsilon }+\varepsilon )^{\frac{1}{2}}|\nabla c_{\varepsilon }|^2 }\\ \le&\displaystyle {C_1^2[\Vert n_0\Vert _{L^1(\varOmega )}+|\varOmega |]+\frac{1}{4}\int _\varOmega |\nabla c_{\varepsilon }|^4 ~~\text {for all}~~ t>0. } \end{aligned} \end{aligned}$$

(1.3.61)

On the other hand, we employ the Gagliardo–Nirenberg inequality to derive that there exists positive constants $C_2,C_3$ and $C_4$ such that

$$ \begin{aligned}&\int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+2\alpha -\frac{1}{2}} \\ =&\displaystyle {\Vert (n_{\varepsilon }+\varepsilon )^{m+\alpha -\frac{3}{4}}\Vert ^{\frac{m+2\alpha -\frac{1}{2}}{m+\alpha -\frac{3}{4}}}_{L^\frac{m+2\alpha -\frac{1}{2}}{m+\alpha -\frac{3}{4}}(\varOmega )}}\\ \le&\displaystyle {C_2\Vert \nabla (n_{\varepsilon }+\varepsilon )^{m+\alpha -\frac{3}{4}}\Vert ^{2\frac{3(2m+4\alpha -3)}{3(4m+4\alpha -3)-2}}_{L^2(\varOmega )} \Vert (n_{\varepsilon }+\varepsilon )^{m+\alpha -\frac{3}{4}}\Vert ^{{\frac{m+2\alpha -\frac{1}{2}}{m+\alpha -\frac{3}{4}}} -2\frac{3(2m+4\alpha -3)}{3(4m+4\alpha -3)-2}}_{L^\frac{1}{m+\alpha -\frac{3}{4}}(\varOmega )}}\\&\displaystyle {+C_2 \Vert (n_{\varepsilon }+\varepsilon )^{m+\alpha -\frac{3}{4}}\Vert ^{{\frac{m+2\alpha -\frac{1}{2}}{m+\alpha -\frac{3}{4}}}}_{L^\frac{1}{m+\alpha -\frac{3}{4}}(\varOmega )}}\\ \le&\displaystyle {C_3( \Vert \nabla (n_{\varepsilon }+\varepsilon )^{m+\alpha -\frac{3}{4}}\Vert ^{2\frac{3(2m+4\alpha -3)}{3(4m+4\alpha -3)-2}}_{L^2(\varOmega )}+1)}\\ =&\displaystyle {C_4\left( \int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{2m+2\alpha -\frac{1}{2}-3} |\nabla n_{\varepsilon }|^2\right) ^{\frac{3(2m+4\alpha -3)}{3(4m+4\alpha -3)-2}}+C_3~~\text {for all}~~ t>0}. \end{aligned} $$

Inserting the above inequality into (1.3.61), one has

$$ \begin{aligned}&\displaystyle {\frac{1}{{m+2\alpha -\frac{1}{2}}}\frac{d}{dt}\Vert n_{\varepsilon }+\varepsilon \Vert ^{{{m+2\alpha -\frac{1}{2}}}}_{L^{{m+2\alpha -\frac{1}{2}}}(\varOmega )}+ C_5\left( \int _\varOmega (n_{\varepsilon }+\varepsilon )^{m+2\alpha -\frac{1}{2}}\right) ^{\frac{3(4m+4\alpha -3)-2}{3(2m+4\alpha -3)}}}\\ \le&\displaystyle {C_6+\frac{1}{4}\int _\varOmega |\nabla c_{\varepsilon }|^4 ~~\text {for all}~~ t>0. } \end{aligned} $$

With the help of (1.3.55) and (1.3.57), we derive that (1.3.60) by Lemma 1.4.

At this position, by the result stated in Lemmas 1.28, 1.24 and 1.16, we have

Lemma 1.29

Let $m+\alpha >\frac{10}{9}.$ Then there exist positive constants $q_0>\frac{3}{2}$ and $C>0$ such that the solution of (1.2.2) satisfies

$$\begin{aligned} \int _{\varOmega }n^{q_0}_\varepsilon (x,t)dx\le C ~~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.62)

Proof

Let

$$ q_0=\left\{ \begin{aligned}&p^*~~\text {if}~~\frac{10}{9}<m+\alpha \le \frac{3}{2}~~~\text {and}~~ 0\le \alpha \le \frac{7}{18},\\&{m+2\alpha }~~\text {if}~~\frac{10}{9}<m+\alpha \le \frac{3}{2}~~~\text {and}~~ \alpha> \frac{7}{18},\\&{m+2\alpha }~~\text {if}~~\frac{3}{2}<m+\alpha \le 2,\\&{m+2\alpha }-\frac{1}{2}~~\text {if}~~m+\alpha >2, \end{aligned}\right. $$

with $p^*$ given in Lemma 1.24. Then it is easy to see that $q_0>\frac{3}{2}$ and thereby (1.3.62) readily follows from Lemmas 1.28, 1.24 and 1.16.

1.3.3 Uniform $L^\infty$-Boundedness of $n_\varepsilon$ as Well as $\nabla c_\varepsilon$ and $u_\varepsilon$

With Lemma 1.29 at hand, further regularity properties of $n_\varepsilon$, $c_\varepsilon$ and $u_\varepsilon$ can now be obtained by essentially rather standard arguments (see the proof of Corollary 3.4 in Winkler 2015b or Lemma 6.1 in Winkler 2018c for example). We firstly use the heat semigroup to obtain the $L^\infty (\varOmega )$-bound for $u_{\varepsilon }$.

Lemma 1.30

Let $m+\alpha >\frac{10}{9}$ and assume that the hypothesis of Theorem 1.1 holds. Then there exists a positive constant C independent of $\varepsilon$ such that, the solution of (1.2.2) satisfies

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot , t)\Vert _{L^{\infty }(\varOmega )}\le \displaystyle {C~ ~~\text {for all}~ t>0.} \end{aligned}$$

(1.3.63)

Proof

Let $h_{\varepsilon }(x,t)=\mathscr {P}[n_{\varepsilon }\nabla \phi ]$. Then by Lemma 1.29, there is $C_{1}>0$ such that for all $t>0$

$$\begin{aligned} \Vert h_{\varepsilon }(\cdot ,t)\Vert _{L^{q_0}(\varOmega )}\le C_{1}. \end{aligned}$$

(1.3.64)

Fixing $r_0$ and $\delta$ with $r_0\in (\frac{3}{2q_0}, 1)$ and $\delta \in (0,1-r_0)$, one can chooses $r_1>\frac{3}{\delta }$ such that $W^{\delta ,r_1}(\varOmega ) \hookrightarrow L^\infty (\varOmega )$. It then follows from the variation-of-constants representation, the Young inequality, the Sobolev embedding theorem and (1.3.64) that

$$ \begin{aligned}&\Vert u_{\varepsilon }(\cdot , t)\Vert _{L^\infty (\varOmega )} \\ \le&\displaystyle {\Vert e^{-tA}u_0\Vert _{L^\infty (\varOmega )} +\int _0^t\Vert A^{r_0}e^{-(t-\tau )A}A^{-r_0}h_{\varepsilon }(\cdot ,\tau )d\tau \Vert _{L^\infty (\varOmega )}d\tau }\\ \le&\displaystyle {\Vert e^{-tA}u_0\Vert _{L^\infty (\varOmega )} +\int _0^t\Vert A^{r_0+\delta }e^{-(t-\tau )A}A^{-r_0}h_{\varepsilon }(\cdot ,\tau )d\tau \Vert _{L^{r_1}(\varOmega )}d\tau }\\ \le&\displaystyle {\Vert e^{-tA}u_0\Vert _{L^\infty (\varOmega )} +C_1\int _0^t(t-\tau )^{-r_0-\delta }e^{-\lambda (t-\tau )}\Vert e^{-(t-\tau )A}A^{-r_0}h_{\varepsilon }(\cdot ,\tau )d\tau \Vert _{L^{r_1}(\varOmega )}d\tau }\\ \le&\displaystyle {\Vert A^\gamma u_0\Vert _{L^2(\varOmega )} +C_{2}\int _0^t(t-\tau )^{-r_0-\delta }e^{-\lambda (t-\tau )}\Vert h_{\varepsilon }(\cdot ,\tau )\Vert _{L^{q_0}(\varOmega )}d\tau }\\ \le&\displaystyle {C_{3}~ \text {for all}~ t>0}. \end{aligned} $$

Here, we have used the fact that $r_0>\frac{3}{2q_0}>\frac{3}{2}(\frac{1}{q_0}-\frac{1}{r_1})$ and

$$\int _0^t(t-\tau )^{-r_0-\delta }e^{-\lambda (t-\tau )}\le \int _0^\infty t^{-r_0-\delta }e^{-\lambda \tau }d\tau <\infty . $$

Lemma 1.31

Assume that the hypothesis of Theorem 1.1 holds. Then there exists a positive constant C independent of $\varepsilon$ such that the solution of (1.2.2) satisfies

$$\begin{aligned} \begin{aligned} \Vert \nabla c_{\varepsilon }(\cdot , t)\Vert _{L^{r_0}(\varOmega )}\le&\displaystyle {C~~~ \text {for all}~~ t>0}\\ \end{aligned} \end{aligned}$$

(1.3.65)

with $3<r_0<\min \{\frac{3q_0}{(3-q_0)_+},4\}$, where $q_0>\frac{3}{2}$ is given by Lemma 1.29.

Proof

Involving the variation-of-constants formula for $c_{\varepsilon }$ and applying $\nabla \cdot u_{\varepsilon }=0$ in $x\in \varOmega , t>0$, we have

$$ c_{\varepsilon }(t)=e^{t(\varDelta -1)}c_0 -\int _{0}^{t}e^{(t-s)(\varDelta -1)}(n_{\varepsilon }(s)c_{\varepsilon }(s)-c_{\varepsilon }(s) -\nabla \cdot (u_{\varepsilon }(s) c_{\varepsilon }(s)) ds, $$

and thus

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert \nabla c_{\varepsilon }(\cdot , t)\Vert _{L^{r_0}(\varOmega )}}\\ \le&\displaystyle {\Vert \nabla e^{t\varDelta } c_0\Vert _{L^{r_0}(\varOmega )}+ \int _{0}^t\Vert \nabla e^{(t-s)\varDelta }[n_{\varepsilon }(s)-1]c_{\varepsilon }(s)\Vert _{L^{r_0}(\varOmega )}ds}\\&\displaystyle {+\int _{0}^t\Vert \nabla e^{(t-s)\varDelta }\nabla \cdot (u_{\varepsilon }(s) c_{\varepsilon }(s))\Vert _{L^{r_0}(\varOmega )}ds~~ \text {for all}~ t>0,}\\ \end{aligned} \end{aligned}$$

(1.3.66)

where $r_0\in (3,\min \{\frac{3q_0}{(3-q_0)_+},4\}).$

Now, we will estimate the terms on the right of (1.3.66) one by one. In view of (1.1.4), there is $C_1>0$ such that

$$\begin{aligned} \Vert \nabla e^{t(\varDelta -1)} c_0\Vert _{L^{r_0}(\varOmega )}\le \displaystyle {C_{1}~~ \text {for all}~ t>0.} \end{aligned}$$

(1.3.67)

Since $q_0>\frac{3}{2}$, it yields

$$-\frac{1}{2}-\frac{3}{2}\left( \frac{1}{q_0}-\frac{1}{r_0}\right) >-1,$$

which together with Lemmas 1.29 and 1.8 implies that for some positive constants $C_2$ and $C_3$,

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^t\Vert \nabla e^{(t-s)(\varDelta -1)}[(n_{\varepsilon }(s)-1)c_{\varepsilon }(s)]\Vert _{L^{r_0}(\varOmega )}ds}\\ \le&\displaystyle {C_{2}\int _{0}^t[1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q_0} -\frac{1}{r_0})}] e^{-\lambda (t-s)}[\Vert n_{\varepsilon }(s)\Vert _{L^{q_0}(\varOmega )}+1]\Vert c_{\varepsilon }(s)\Vert _{L^{\infty }(\varOmega )}ds}\\ \le&\displaystyle {C_{3}~~~\text {for all}~~ t>0.} \end{aligned} \end{aligned}$$

(1.3.68)

Finally, we choose $\iota =\frac{1}{3}$ satisfying $\frac{1}{2} + \frac{3}{2}(\frac{1}{\infty }-\frac{1}{6}) <\frac{1}{3}$ and $\tilde{\kappa }=\frac{1}{12}\in (0, \frac{1}{6})$. In view of Hölder’s inequality, we derive from Lemma 1.30 that there exist positive constants $C_{i}$, $i=4,\cdots ,8$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^t\Vert \nabla e^{(t-s)(\varDelta -1)}\nabla \cdot (u_{\varepsilon }(s) c_{\varepsilon }(s))\Vert _{L^{r_0}(\varOmega )}ds}\\ \le&\displaystyle {C_4\int _{0}^t\Vert e^{(t-s)(\varDelta -1)}\nabla \cdot (u_{\varepsilon }(s) c_{\varepsilon }(s))\Vert _{W^{1,r_0}(\varOmega )}ds}\\ \le&\displaystyle {C_{5}\int _{0}^t\Vert (-\varDelta +1)^\iota e^{(t-s)(\varDelta -1)}\nabla \cdot (u_\varepsilon (s) c_\varepsilon (s))\Vert _{L^{6}(\varOmega )}ds}\\ \le&\displaystyle {C_{6}\int _{0}^t(t-s)^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\tilde{\lambda }(t-s)} \Vert u_\varepsilon (s) c_\varepsilon (s)\Vert _{L^{\infty }(\varOmega )}ds}\\ \le&\displaystyle {C_{7}\int _{0}^t(t-s)^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\tilde{\lambda }(t-s)} \Vert u_\varepsilon (s)\Vert _{L^{\infty }(\varOmega )}\Vert c_\varepsilon (s)\Vert _{L^{\infty }(\varOmega )}ds}\\ \le&\displaystyle {C_{8}~ ~~\text {for all}~~ t>0.}\\ \end{aligned} \end{aligned}$$

(1.3.69)

Here, we have used the fact that

$$ \displaystyle \int _{0}^t(t-s)^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\lambda (t-s)}ds \le \displaystyle {\int _{0}^{\infty }\sigma ^{-\iota -\frac{1}{2}-\tilde{\kappa }} e^{-\lambda \sigma }d\sigma <+\infty .} $$

Combining with (1.3.66)–(1.3.69), we arrive at (1.3.65).

From the regularity property of solutions obtained above, we can infer the higher regularity about $n_\varepsilon$.

Lemma 1.32

Assuming that $m+\alpha >\frac{10}{9}.$ Then for all $p>2$, there exists $C>0$ such that

$$\begin{aligned} \Vert {n}_\varepsilon (\cdot , t)+\varepsilon \Vert _{L^p(\varOmega )}\le C~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.70)

Proof

Recalling Lemmas 1.14 and by (1.31), we have

$$\begin{aligned}&\displaystyle \frac{1}{{p}}\displaystyle \frac{d}{dt}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ \displaystyle \frac{{m}(p-1)}{2}\displaystyle \int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{m+p-3} |\nabla n_{\varepsilon }|^2 \nonumber \\ \le&\displaystyle {C_1\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p+1-m}(1+n_{\varepsilon })^{-2\alpha }|\nabla c_{\varepsilon }|^2} \nonumber \\ \le&\displaystyle {C_1\left( \int _\varOmega (n_{\varepsilon }+\varepsilon )^{3(p+1-m-2\alpha )}\right) ^{\frac{1}{3}} \left( \int _\varOmega |\nabla c_{\varepsilon }|^{3}\right) ^{\frac{2}{3}}} \nonumber \\ \le&\displaystyle {C_2\left( \int _\varOmega (n_{\varepsilon }+\varepsilon )^{3(p+1-m-2\alpha )}\right) ^{\frac{1}{3}}~~\text {for all}~~ t>0} \nonumber \end{aligned}$$

for constants $C_1>0$ and $k_2>0$. By the Gagliardo–Nirenberg inequality, there exist positive constants $C_i> 0$, $(i=3,4,5,6)$ fulfilling

$$ \begin{aligned}&\displaystyle {\left( \int _\varOmega (n_{\varepsilon }+\varepsilon )^{3(p+1-m-2\alpha )}\right) ^{\frac{1}{3}}}\\ =&\displaystyle {\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{\frac{2(p+1-m-2\alpha )}{m+p-1}}_{L^\frac{6(p+1-m)}{m+p-1}(\varOmega )}}\\ \le&\displaystyle {C_3 \Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{2\frac{3p-3m+2-6\alpha }{3m+3p-4}}_{L^2(\varOmega )} \Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{\frac{2(p+1-m)}{m+p-1}-2\frac{3p-3m+2-6\alpha }{3m+3p-4}}_{L^\frac{2}{m+p-1}(\varOmega )}}\\&\displaystyle {+C_3 \Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{\frac{2(p+1-m)}{m+p-1}}_{L^\frac{2}{m+p-1}(\varOmega )}}\\ \le&\displaystyle {C_4( \Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{2\frac{3p-3m+2-6\alpha }{3m+3p-4}}_{L^2(\varOmega )}+1)~~\text {for all}~~ t>0}\\ \end{aligned} $$

and

$$ \begin{aligned}&\displaystyle \int _\varOmega (n_{\varepsilon }+\varepsilon )^{p}\\ =&\displaystyle {\Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{\frac{2p}{m+p-1}}_{L^\frac{2p}{m+p-1}(\varOmega )}}\\ \le&\displaystyle {C_5\Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{2\frac{3p-3}{3m+3p-4}}_{L^2(\varOmega )} \Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{\frac{2p}{m+p-1}-2\frac{3p-3}{3m+3p-4}}_{L^\frac{2}{m+p-1}(\varOmega )}}\\&\displaystyle {+C_4 \Vert (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{\frac{2p}{m+p-1}}_{L^\frac{2}{m+p-1}(\varOmega )}}\\ \le&\displaystyle {C_6( \Vert \nabla (n_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^{2\frac{3p-3}{3m+3p-4}}_{L^2(\varOmega )}+1)~~\text {for all}~~ t>0.}\\ \end{aligned} $$

With the help of $m+\alpha >\frac{10}{9}$, we have $\frac{3p-3m+2-6\alpha }{3m+3p-4}<1$ and hence obtain that for some constant $C_7>0$

$$ \begin{aligned} \displaystyle \frac{1}{{p}}\displaystyle \frac{d}{dt}\Vert {n_{\varepsilon }}+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ C_7\left( \int _\varOmega (n_{\varepsilon }+\varepsilon )^{p}\right) ^{\frac{3m+3p-4}{3p-3}} \le&\displaystyle {C_7~~\text {for all}~~ t>0.} \end{aligned} $$

Therefore, (1.3.70) follows from the application of Lemma 1.4.

By applying the general semigroup estimates, the standard parabolic regularity arguments and a Moser-type iteration (see, e.g., Lemma A.1 of Tao and Winkler 2012a), we can now establish the existence of global bounded classical solutions to the regularized system (1.2.2).

Proposition 1.1

Let $m+\alpha >\frac{10}{9}$. Then there exists $C > 0$ independent of $\varepsilon \in (0, 1)$ such that

$$\begin{aligned} \Vert n_\varepsilon (\cdot , t)\Vert _{L^\infty (\varOmega )} +\Vert c_\varepsilon (\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}+\Vert A^\gamma u_\varepsilon (\cdot , t)\Vert _{L^{2}(\varOmega )}\le C~~~\text {for all}~~ t>0. \end{aligned}$$

(1.3.71)

Proof

Let $h_{\varepsilon }(x,t)=\mathscr {P}[n_{\varepsilon }\nabla \phi ]$. Then by (1.3.70), there is $C_{1}>0$ such that

$$ \Vert h_{\varepsilon }(\cdot ,t)\Vert _{L^{2}(\varOmega )}\le C_{1} ~\text {for all}~ t>0. $$

So combining the known smoothing properties of the Stokes semigroup (see Giga 1986) with (1.1.4), there are positive constants $C_{2}$ and $C_{3}$ such that

$$ \begin{aligned} \Vert A^\gamma u_{\varepsilon }(\cdot , t)\Vert _{L^2(\varOmega )}\le&\displaystyle {\Vert A^\gamma e^{-tA}u_0\Vert _{L^2(\varOmega )} +\int _0^t\Vert A^\gamma e^{-(t-\tau )A}h_{\varepsilon }(\cdot ,\tau )d\tau \Vert _{L^2(\varOmega )}d\tau } \\ \le&\displaystyle {\Vert A^\gamma u_0\Vert _{L^2(\varOmega )} +C_{2}\int _0^t(t-\tau )^{-\gamma }e^{-\lambda (t-\tau )}\Vert h_{\varepsilon }(\cdot ,\tau )\Vert _{L^{2}(\varOmega )}d\tau } \\ \le&\displaystyle {C_{3}~ \text {for all}~ t>0.} \end{aligned} $$

Next, we rewrite the variation-of-constants formula for $c_{\varepsilon }$ in the form

$$ c_\varepsilon (\cdot , t) = e^{t(\varDelta -1) }c_0 +\int _0^te^{(t-s)(\varDelta -1)} (c_\varepsilon -n_\varepsilon c_\varepsilon -u_{\varepsilon } \cdot \nabla c_{\varepsilon })(\cdot ,s)ds~~\text {for all}~~ t>0. $$

Due to $3<r_0<\min \{\frac{3q_0}{(3-q_0)_+},4\}$ (see Lemma 1.31), one can pick $\theta \in (\frac{1}{2}+\frac{3}{2r_0},1)$ and thereby the domain of the fractional power $D((-\varDelta + 1)^\theta )\hookrightarrow W^{1,\infty }(\varOmega )$ (see Winkler 2010). Hence, in view of $L^p$-$L^q$ estimates associated heat semigroup, Lemma 1.31 as well as (1.1.4), we conclude that there exist positive constants $\lambda _1,C_4$ as well as $C_5$ and $C_6$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert \nabla c_\varepsilon (\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}} \\ \le&\displaystyle {C_{4}e^{-\lambda _1 t}\Vert \nabla c_0\Vert _{L^{{\infty }}(\varOmega )}} \\&+\displaystyle {\int _{0}^t(t-s)^{-\theta }e^{-\lambda _1(t-s)}\Vert (c_\varepsilon -n_\varepsilon c_\varepsilon -u_{\varepsilon } \cdot \nabla c_{\varepsilon })(s)\Vert _{L^{r_0}(\varOmega )}ds} \\ \le&\displaystyle {C_{5}+C_{5}\int _{0}^t(t-s)^{-\theta }e^{-\lambda _1(t-s)}ds} \\&\displaystyle {+C_{5}\int _{0}^t(t-s)^{-\theta }e^{-\lambda _1(t-s)}[\Vert n_\varepsilon (\cdot ,s)\Vert _{L^{r_0}(\varOmega )} +\Vert \nabla c_{\varepsilon }(\cdot ,s)\Vert _{L^{r_0}(\varOmega )}]ds} \\ \le&\displaystyle {C_{6}~~ \text {for all}~~ t\in (0,\infty ).} \end{aligned} \end{aligned}$$

(1.3.72)

Finally, we rewrite the first equation of (1.2.2) as

$$\begin{aligned} n_{\varepsilon t}=\varDelta (n_{\varepsilon }+\varepsilon )^m -\nabla \cdot (n_{\varepsilon }u_{\varepsilon } +n_{\varepsilon }F_{\varepsilon }(n_{\varepsilon })S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_{\varepsilon }), \end{aligned}$$

(1.3.73)

Hence, in view of (1.3.72) and using the outcome of Lemma 1.32 with suitably large p as a starting point, we may invoke Lemma A.1 in Tao and Winkler (2012a) which by means of a Moser-type iteration applied to (1.3.73) and establish

$$\begin{aligned} \begin{aligned} \Vert n_{\varepsilon }(\cdot , t)\Vert _{L^{\infty }(\varOmega )}\le&\displaystyle {C_7~~~ \text {for all}~~~ t>0}\\ \end{aligned} \end{aligned}$$

(1.3.74)

with some positive constant $C_7$ independent of $\varepsilon$.

To achieve the convergence result, we still need the following further regularity estimate. With the help of Proposition 1.1, we can straightforwardly deduce the uniform Hölder properties of $c_\varepsilon$ as well as $\nabla c_\varepsilon$ and $u_\varepsilon$ by using the standard parabolic regularity property and the standard semigroup estimation techniques.

Lemma 1.33

Let $m+\alpha >\frac{10}{9}$. Then one can find $\mu \in (0, 1)$ such that for some $C > 0$,

$$ \Vert c_\varepsilon (\cdot ,t)\Vert _{C^{\mu ,\frac{\mu }{2}}(\varOmega \times [t,t+1])} \le C ~~\text {for all}~~ t\in (0,\infty ) $$

as well as

$$ \Vert u_\varepsilon (\cdot ,t)\Vert _{C^{\mu ,\frac{\mu }{2}}(\varOmega \times [t,t+1])} \le C ~~\text {for all}~~ t\in (0,\infty ), $$

and for any $\tau > 0$, there exists $C(\tau ) > 0$ fulfilling

$$ \Vert \nabla c_\varepsilon (\cdot ,t)\Vert _{C^{\mu ,\frac{\mu }{2}}(\varOmega \times [t,t+1])} \le C ~~\text {for all}~~ t\in (\tau ,\infty ). $$

Proof

Based on the uniform boundedness of $\{(n_\varepsilon , c_\varepsilon , u_\varepsilon )\}_{\varepsilon \in (0,1)}$ as claimed in Proposition 1.1 and the assumptions on $\phi$, we conclude the desired estimates by applying the standard parabolic regularity theory (see, e.g., Ladyzenskaja et al. 1968) and some standard semigroup estimation techniques, which is omitted here.

Unlike $c_\varepsilon$ and $u_\varepsilon$, we are not able to attain the Hölder regularity for $n_\varepsilon$ due to the presence of nonlinear diffusion. We now make full use of the a priori bounds derived so far to obtain the boundedness property of the time derivatives of certain powers of $n_\varepsilon$ and spatio-temporal integrability property of $\int _{0}^\infty \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2$, which plays a key role in deriving strong compactness properties for $n_\varepsilon$. Let us provide the following spatio-temporal estimates at first.

Lemma 1.34

Let $m+\alpha >\frac{10}{9}$. Then there exists a positive constant C such that for any $\varepsilon \in (0, 1)$

$$\begin{aligned} \int _{0}^\infty \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2\le C ~~~\text {for all}~~p > 1~~\text {and}~~ p\ge m+2\alpha -1. \end{aligned}$$

(1.3.75)

Proof

In light of Proposition 1.1, there exists $C_1> 0$ such that for all $\varepsilon \in (0,1)$, $n_\varepsilon \le C_1$ in $\varOmega \times (0,\infty ).$ For any $p\ge m+2\alpha -1$ and $p>1$, using Proposition 1.1, we can thereby estimate the integral on the right of (1.3.26) according to

$$ \begin{aligned}&\displaystyle {\frac{1}{{p}}\Vert n_{\varepsilon }(\cdot ,t)+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}+ \frac{{m}(p-1)}{2}\int _0^t\int _{\varOmega }(n_{\varepsilon }+{\varepsilon })^{m+p-3} |\nabla n_{\varepsilon }|^2} \\ \le&\displaystyle {\frac{(p-1)C_S^2}{2{m}}\int _0^t\int _\varOmega (n_{\varepsilon }+\varepsilon )^{p+1-m-2\alpha }|\nabla c_{\varepsilon }|^2 +\frac{1}{{p}}\Vert n_{0}+\varepsilon \Vert ^{{{p}}}_{L^{{p}}(\varOmega )}} \\ \le&\displaystyle {\frac{(p-1)C_S^2}{2{m}}(C_1+1)^{p+1-m-2\alpha }\int _0^\infty \int _\varOmega |\nabla c_{\varepsilon }|^2+\frac{1}{{p}}\Vert n_{0}+1\Vert ^{{{p}}}_{L^{{p}}(\varOmega )}} \\ \le&\displaystyle {\frac{(p-1)C_S^2}{4{m}}(C_1+1)^{p+1-m-2\alpha }\int _\varOmega c_0^2+\frac{1}{{p}}\Vert n_{0}+1\Vert ^{{{p}}}_{L^{{p}}(\varOmega )}~~\text {for all}~~t\in (0,\infty ),} \end{aligned} $$

which immediately leads to our conclusion.

In order to pass to the limit in system (1.2.2) by compactness argument, we intend to supplement Proposition 1.1 with an appropriate boundedness property of the time derivatives of $n_\varepsilon$.

Lemma 1.35

Let $m+\alpha >\frac{10}{9}$. Then one can find $C>0$ such that for any $\varepsilon \in (0, 1)$

$$\begin{aligned} \Vert \partial _tn_\varepsilon (\cdot ,t)\Vert _{(W^{2,2}_0(\varOmega ))^*} \le C ~~\text {for all}~~ t\in (0,\infty ). \end{aligned}$$

(1.3.76)

In particular,

$$\begin{aligned} \Vert n_\varepsilon (\cdot ,t)-n_\varepsilon (\cdot ,s)\Vert _{(W^{2,2}_0(\varOmega ))^*} \le C|t-s| ~~\text {for all}~~ t\ge 0,s\ge 0~~~\text {and}~~\varepsilon \in (0,1). \end{aligned}$$

(1.3.77)

Moreover, let $\varsigma > m$ and $\varsigma \ge 2(m - 1 )$. Then for all $T > 0$ and $\varepsilon \in (0,1)$, there exists a positive constant C(T) such that

$$\begin{aligned} \int _0^T\Vert \partial _t(n_\varepsilon +\varepsilon )^\varsigma (\cdot ,t)\Vert _{(W^{3,2}_0(\varOmega ))^*}dt \le C(T) ~~\text {for all}~~ \varepsilon \in (0,1). \end{aligned}$$

(1.3.78)

Proof

To estimate the integrals on the right of (1.3.80) below appropriately, we first apply Proposition 1.1 to find $C_1$ such that

$$\begin{aligned} (n_\varepsilon +\varepsilon )^m\le C_1,~~ n_\varepsilon \le C_1~~~\text {as well as}~~|\nabla c_\varepsilon | \le C_1~~\text {and}~~|u_\varepsilon | \le C_1 ~~\text {in}~~ \varOmega \times (0,\infty ). \end{aligned}$$

(1.3.79)

For any fixed $\psi \in C_0^{\infty }(\varOmega )$, we multiply the first equation in (1.2.2) by $(n_\varepsilon +\varepsilon )^{\varsigma -1}\psi$ and then get

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{\varsigma }\int _{\varOmega }\partial _{t}(n_\varepsilon +\varepsilon )^{\varsigma }(\cdot ,t)\cdot \psi \\ =&\displaystyle {\int _{\varOmega }(n_\varepsilon +\varepsilon )^{\varsigma -1}\left[ \varDelta (n_{\varepsilon }+\varepsilon )^m -\nabla \cdot (n_{\varepsilon }S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon })-u_{\varepsilon }\cdot \nabla n_{\varepsilon }\right] \cdot \psi } \\ =&\displaystyle {-(\varsigma -1)m\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma -2}(n_\varepsilon +\varepsilon )^{m-1}|{\nabla } {n}_{\varepsilon }|^2\psi }\\&\displaystyle {-m\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma -1}(n_\varepsilon +\varepsilon )^{m-1}{\nabla } {n}_{\varepsilon }\cdot \nabla \psi } \\&+\displaystyle {(\varsigma -1)\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma -1} \nabla n_\varepsilon \cdot (S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_{\varepsilon })\psi } \\&\displaystyle {+\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma } S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }\cdot \nabla \psi }\\&+\displaystyle {\frac{1}{\varsigma }\int _\varOmega (n_\varepsilon +\varepsilon )^\varsigma u_\varepsilon \cdot \nabla \psi ~~\text {for all}~~t\in (0,\infty ).}\\ \end{aligned} \end{aligned}$$

(1.3.80)

In what follows, we shall estimate the right of the above equality appropriately by (1.3.79). Indeed, since $\varsigma > m$ and $\varsigma \ge 2(m+\alpha - 1 )$, the number $p := \varsigma - m + 1$ satisfies $p > 1$ and $p\ge m+2\alpha -1$, so that, (1.3.75) becomes applicable so as to yield $C_3> 0$ fulfilling

$$ \int _{0}^\infty \int _{\varOmega }(n_\varepsilon +\varepsilon )^{\varsigma -2} |\nabla n_{\varepsilon }|^2 =\int _{0}^\infty \int _{\varOmega }(n_{\varepsilon }+\varepsilon )^{m+p-3} |\nabla n_{\varepsilon }|^2\le C_3 $$

for some positive constant $C_3$. Now, applying (1.3.79), we conclude from the Young inequality that

$$\begin{aligned} \begin{aligned}&\displaystyle {-(\varsigma -1)m\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma -2}(n_\varepsilon +\varepsilon )^{m-1}|{\nabla } {n}_{\varepsilon }|^2\psi } \\&\displaystyle {-m\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma -1}(n_\varepsilon +\varepsilon )^{m-1}{\nabla } {n}_{\varepsilon }\cdot \nabla \psi }\\&+\displaystyle {(\varsigma -1)\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma -1} \nabla n_\varepsilon \cdot (S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_{\varepsilon })\psi } \\&\displaystyle {+\int _\varOmega (n_\varepsilon +\varepsilon )^{\varsigma } S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }\cdot \nabla \psi } +\displaystyle {\frac{1}{\varsigma }\int _\varOmega (n_\varepsilon +\varepsilon )^\varsigma u_\varepsilon \cdot \nabla \psi }\\ \le&\displaystyle {C_4(\varsigma -1)\int _{\varOmega }(n_\varepsilon +\varepsilon )^{\varsigma -2} |\nabla n_{\varepsilon }|^2\Vert \psi \Vert _{L^\infty (\varOmega )}} \\&\displaystyle {+C_5\int _\varOmega [(n_\varepsilon +\varepsilon )^{\varsigma -2}|\nabla n_{\varepsilon }|^2+(C_1+1)^{\varsigma -1}]\Vert \nabla \psi \Vert _{L^\infty (\varOmega )} }\\&+\displaystyle {C_6\int _\varOmega [(n_\varepsilon +\varepsilon )^{\varsigma -2}|\nabla n_{\varepsilon }|^2+C_1^{\varsigma }]\Vert \psi \Vert _{L^\infty (\varOmega )}} \\&\displaystyle {+C_1^{\varsigma +1}C_S|\varOmega |\Vert \nabla \psi \Vert _{L^\infty (\varOmega )}+\frac{1}{\varsigma }C_1^{\varsigma +1}|\varOmega | \Vert \psi \Vert _{L^\infty (\varOmega )}} \end{aligned} \end{aligned}$$

(1.3.81)

with some positive constants $C_4$ as well as $C_5$ and $C_6$. Inserting (1.3.81) into (1.3.80), we derive that there is $C_7>0$ such that for all $t > 0$ and any $\varepsilon \in (0,1)$,

$$ \begin{aligned} \displaystyle {|\int _{\varOmega }\partial _{t}(n_\varepsilon +\varepsilon )^{\varsigma }(\cdot ,t)\cdot \psi |\le C_7( \int _{\varOmega }(n_\varepsilon +\varepsilon )^{\varsigma -2} |\nabla n_{\varepsilon }|^2+1)\Vert \psi \Vert _{W^{1,\infty }(\varOmega )}.} \end{aligned} $$

As in the three-dimensional space, we have $W^{3,2}_{0}(\varOmega ) \hookrightarrow W^{1,\infty }(\varOmega )$. Collecting the above inequalities, we infer the existence of $C_8 > 0$ such that for any $\varepsilon \in (0,1)$,

$$ \displaystyle {\Vert \partial _t(n_\varepsilon +\varepsilon )^\varsigma (\cdot ,t)\Vert _{(W^{3,2}_0(\varOmega ))^*}\le C_8( \int _{\varOmega }(n_\varepsilon +\varepsilon )^{\varsigma -2} |\nabla n_{\varepsilon }|^2+1)~~\text {for all}~~t\in (0,\infty ).} $$

Therefore, we obtain the desired estimate (1.3.78).

Testing the first equation in (1.2.2) by an arbitrary $\varphi \in C_0^{\infty }(\varOmega )$, we have

$$ \begin{aligned} \displaystyle \int _{\varOmega }n_{\varepsilon ,t}(\cdot ,t)\cdot \varphi =&\displaystyle {\int _{\varOmega } \left[ \varDelta (n_{\varepsilon }+\varepsilon )^m-\nabla \cdot (n_{\varepsilon }S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon })-u_{\varepsilon }\cdot \nabla n_{\varepsilon }\right] \cdot \varphi } \\ =&\displaystyle {\int _\varOmega (n_{\varepsilon }+\varepsilon )^m\varDelta \varphi +\int _\varOmega n_{\varepsilon } S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\nabla c_{\varepsilon }\cdot \nabla \varphi } +\displaystyle {\int _\varOmega n_{\varepsilon }u_\varepsilon \cdot \nabla \varphi }\\ \end{aligned} $$

for all $t\in (0,\infty )$. Then combining this with (1.3.79) as well as (1.1.3), we get

$$ \displaystyle |\int _{\varOmega }n_{\varepsilon ,t}(\cdot ,t)\cdot \varphi | \le \displaystyle {C_9[\int _{\varOmega }|\varDelta \varphi |+\int _{\varOmega }|\nabla \varphi |]~~\text {in}~~ \varOmega \times (0,\infty )} $$

for all $\varepsilon \in (0, 1)$ with some positive constant $C_9$, which establishes implies (1.3.76) and thus also (1.3.77).

1.3.4 Global Boundedness of Weak Solutions

The a-priori estimates achieved so far allow us to construct weak solutions by compactness arguments. To this end, let us define what a weak solution is supposed to be.

Definition 1.1

(Weak solutions) By a global weak solution of (1.1.1), we mean a triple (n, c, u) of functions

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

such that $n\ge 0$ and $c\ge 0$ a.e. in $\varOmega \times (0, \infty )$,

$$ \begin{aligned}&nc,n^m~~\in L^1_{loc}(\bar{\varOmega }\times [0, \infty )),~~~u\otimes u \in L^1_{loc}(\bar{\varOmega }\times [0, \infty );\mathbb {R}^{3\times 3}),~~\text {and}\\&~~ nS(x,n,c)\nabla c,~~~cu~~ \text {and}~~~ nu~~ \text {belong to}~~ L^1_{loc}(\bar{\varOmega }\times [0, \infty );\mathbb {R}^{3}),\\ \end{aligned} $$

$\nabla \cdot u = 0$ a.e. in $\varOmega \times (0, \infty )$, and

$$ \begin{aligned}&\displaystyle {-\int _0^{T}\int _{\varOmega }n\varphi _t-\int _{\varOmega }n_0\varphi (\cdot ,0) } \\ =&\displaystyle { \int _0^T\int _{\varOmega }n^m\varDelta \varphi +\int _0^T\int _{\varOmega }n(S(x,n,c)\cdot \nabla c)\cdot \nabla \varphi }+\displaystyle {\int _0^T\int _{\varOmega }nu\cdot \nabla \varphi } \end{aligned} $$

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, \infty ))$ as well as

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

for any $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, \infty ))$ and

$$ \begin{aligned} \displaystyle {-\int _0^{T}\int _{\varOmega }u\varphi _t-\int _{\varOmega }u_0\varphi (\cdot ,0) }=&\displaystyle {- \int _0^T\int _{\varOmega }\nabla u\cdot \nabla \varphi - \int _0^T\int _{\varOmega }n\nabla \phi \cdot \varphi } \end{aligned} $$

for any $\varphi \in C_0^{\infty } (\varOmega \times [0, \infty );\mathbb {R}^3)$ fulfilling $\nabla \cdot \varphi \equiv 0$.

The Proof of Theorem 1.1 We first give a series of convergence results. According to Lemma 1.33, the Arzelà-Ascoli theorem and a standard extraction procedure, we can find a sequence $(\varepsilon _j)_{j\in \mathbb {N}} \subseteq (0,1)$ with $\varepsilon _j\searrow 0$ as $j\rightarrow \infty$ such that

$$\begin{aligned} c_{\varepsilon _j}\rightarrow c ~~\text {in}~~ C^0_{loc}(\bar{\varOmega }\times [0,\infty )), \end{aligned}$$

(1.3.82)

$$\begin{aligned} \nabla c_{\varepsilon _j}\rightarrow \nabla c ~~\text {in}~~ C^0_{loc}(\bar{\varOmega }\times [0,\infty )), \end{aligned}$$

(1.3.83)

and

$$\begin{aligned} u_{\varepsilon _j}\rightarrow u ~~\text {in}~~ C^0_{loc}(\bar{\varOmega }\times (0,\infty )) \end{aligned}$$

(1.3.84)

hold with some limit functions c and u belonging to the indicated spaces. On the other hand, Proposition 1.1 ensures the existence of a subsequence such that

$$\begin{aligned} \nabla c_{\varepsilon _j}\rightharpoonup \nabla c ~~\text {weakly star in}~~ L^{\infty }(\varOmega \times (0,\infty )), \end{aligned}$$

(1.3.85)

$$\begin{aligned} D u_{\varepsilon _j}\rightharpoonup Du ~~\text {weakly star in}~~L^{\infty }(\varOmega \times [0,\infty )), \end{aligned}$$

(1.3.86)

and

$$\begin{aligned} n_{\varepsilon _j}\rightharpoonup n ~~\text {weakly star in}~~ L^\infty (\varOmega \times (0,\infty )) \end{aligned}$$

(1.3.87)

hold for some $n \in L^\infty (\varOmega \times (0,\infty ))$.

Fix $\zeta >m-1$. Then Lemmas 1.34 and 1.35 assert that for any $T >0$,

$$ ((n_{\varepsilon }+\varepsilon )^\varsigma )_{\varepsilon \in (0,1)}~~\text {is bounded in}~~L^2((0, T);W^{1,2}(\varOmega )) $$

and

$$ (\partial _{t}(n_\varepsilon +\varepsilon )^\varsigma )_{\varepsilon \in (0,1)}~~\text {is bounded in}~~L^1((0, T); (W^{3,2}_0(\varOmega ))^*) $$

respectively. So the embedding $W^{1,2}(\varOmega )\hookrightarrow \hookrightarrow L^2 (\varOmega )\hookrightarrow (W^{3,2}_0(\varOmega ))^*$ and the Aubin–Lions compactness lemma yield that $(n_{\varepsilon }+\varepsilon )^\varsigma _{\varepsilon \in (0,1)}$ is a relatively compact subset of the space $L^2 (\varOmega \times (0,T))$. This in conjunction with the Egorov theorem gives that for some subsequence of $\varepsilon =\varepsilon _j$,

$$ (n_{\varepsilon }+\varepsilon )^\varsigma \rightarrow n^\varsigma ~~~\text {strongly in}~~L^2 (\varOmega \times (0,T)) $$

and hence

$$ (n_{\varepsilon }+\varepsilon )\rightarrow z~~~\text {a.e.}~~ \text {in}~~ \varOmega \times (0,T), $$

for some nonnegative measurable $z: \varOmega \times (0,T) \rightarrow \mathbb {R}$. This combined with the Egorov theorem, then we can see that $z= n$, and thereby

$$\begin{aligned} n_\varepsilon \rightarrow n ~~\text {a.e.}~~ \text {in}~~ \varOmega ~~~\text {for all}~~ (0,\infty )\setminus N. \end{aligned}$$

(1.3.88)

Next, noticing that $L^\infty (\varOmega )\hookrightarrow (W^{2,2}_0(\varOmega ))^*$ is compact, in view of Proposition 1.1 and Lemma 1.35, we can use the Arzelà-Ascoli theorem again to assert

$$\begin{aligned} n_\varepsilon \rightarrow n ~~\text {in}~~ C^0_{loc}([0,\infty ); (W^{2,2}_0 (\varOmega ))^*). \end{aligned}$$

(1.3.89)

With the help of (1.3.89) and the fact that $\Vert n\Vert _{L^\infty (\varOmega \times (0,\infty ))}$ is finite, we can derive

$$\begin{aligned} n\in C^0_{\omega -*}([0,\infty ); L^\infty (\varOmega )) \end{aligned}$$

(1.3.90)

by using the similar methods in the proof of Lemma 4.1 in Winkler (2015b).

Combining (1.3.83) with (1.3.88), noticing the definition of $S_\varepsilon$, we may further infer that

$$ n_\varepsilon S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_\varepsilon \rightarrow n S(x, n, c)\cdot \nabla c ~~\text { a.e. in}~~ \varOmega \times (0,\infty )~~~\text {as}~~\varepsilon :=\varepsilon _j\searrow 0. $$

Then we may use the dominated convergence theorem, along with a subsequence (still denoted by $\{\varepsilon _j\}_{j=1}^\infty )$, we derive that

$$\begin{aligned} n_\varepsilon S_\varepsilon (x, n_{\varepsilon }, c_{\varepsilon })\cdot \nabla c_\varepsilon \rightarrow n S(x, n, c)\cdot \nabla c ~~\text {strongly in}~~ L^{2}_{loc}(\bar{\varOmega }\times [0,\infty ))~~~\text {as}~~\varepsilon :=\varepsilon _j\searrow 0. \end{aligned}$$

(1.3.91)

In the following, we shall show that the triple (n, c, u) is exactly a global weak solution to system (1.1.1). Indeed, multiplying the first equation in (1.2.2) by $\varphi \in C_0^{\infty } (\bar{\varOmega }\times [0, \infty ))$, integrating by parts, we obtain

$$ \begin{aligned}&\displaystyle {-\int _0^{\infty }\int _{\varOmega }n_\varepsilon \varphi _t-\int _{\varOmega }n_0\varphi (\cdot ,0) } \\ =&\displaystyle { \int _0^\infty \int _{\varOmega }(n_\varepsilon +\varepsilon )^m\varDelta \varphi +\int _0^\infty \int _{\varOmega }n_\varepsilon (S_\varepsilon (x,n_\varepsilon ,c_\varepsilon )\cdot \nabla c_\varepsilon )\cdot \nabla \varphi }\\&+\displaystyle {\int _0^\infty \int _{\varOmega }n_\varepsilon u_\varepsilon \cdot \nabla \varphi }. \end{aligned} $$

In view of (1.3.89), (1.3.91) as well as (1.3.84), we conclude from the dominated convergence theorem that

$$ \begin{aligned}&\displaystyle {-\int _0^{\infty }\int _{\varOmega }n\varphi _t-\int _{\varOmega }n_0\varphi (\cdot ,0) } \\ =&\displaystyle { \int _0^\infty \int _{\varOmega }n^m\varDelta \varphi +\int _0^\infty \int _{\varOmega }n(S(x,n,c)\cdot \nabla c)\cdot \nabla \varphi }\\&+\displaystyle {\int _0^\infty \int _{\varOmega }nu\cdot \nabla \varphi }. \end{aligned} $$

Next, multiplying the second equation and the third equation in (1.2.2) by $\varphi \in C^\infty _0(\varOmega \times [0,\infty ))$ and $\psi \in C_0^{\infty } (\bar{\varOmega }\times [0, \infty );\mathbb {R}^3)$, respectively, then with the help of (1.3.85)–(1.3.86) and by a limit procedure, we also derive that

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

and

$$ \displaystyle {-\int _0^{\infty }\int _{\varOmega }u\varphi _t-\int _{\varOmega }u_0\varphi (\cdot ,0)} =\displaystyle {- \int _0^\infty \int _{\varOmega }\nabla u\cdot \nabla \varphi - \int _0^\infty \int _{\varOmega }n\nabla \phi \cdot \varphi } $$

in a completed similar manner. This means that (n, c, u) is a weak solution of (1.1.1). The convergence properties in (1.3.82)–(1.3.89) lead to the stated boundedness of global weak solutions thereof, and thus complete the proof of Theorem 1.1.

1.4 Asymptotic Profile of Solution to a Chemotaxis–Fluid System with Singular Sensitivity

1.4.1 Basic a Priori Bounds

In order to derive some essential estimates, it would be more convenient to deal with a nonsingular chemotaxis term of the form $\nabla \cdot (n \nabla w)$ instead of $\nabla \cdot (\frac{n}{c}\nabla c)$ in (1.1.5). To this end, we employ the following transformation as in Lankeit and Lankeit (2019a), Lankeit (2017), Winkler (2016a): $w:= -\ln (\frac{c}{\Vert c_0\Vert _{L^\infty (\varOmega )}}),$ whereupon $0\le w\in C^0(\bar{\varOmega }\times (0,\infty ))\cap C^{2,1}(\bar{\varOmega }\times (0,\infty ))$, and the problem (1.1.5), (1.1.8), (1.1.9) transforms to

$$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n =\triangle n+\chi \nabla \cdot ( n\nabla w)+n(r- \mu n),&x\in \varOmega , t>0,\\&w_t+u\cdot \nabla w =\triangle w- |\nabla w|^2 +n,&x\in \varOmega , t>0,\\&u_t+ (u\cdot \nabla ) u=\varDelta u+\nabla P+n\nabla \phi ,&x\in \varOmega , t>0,\\&\nabla \cdot u=0,&x\in \varOmega , t>0,\\&\nabla n\cdot \nu =\nabla w\cdot \nu =0, \quad u=0,&x\in \partial \varOmega , t>0, \\&n(x,0)=n_0(x),~ w(x,0)=-\ln (\frac{c_0(x)}{\Vert c_0\Vert _{L^\infty (\varOmega )}}), ~~ u=u_0(x),&x\in \varOmega . \end{aligned}\right. \end{aligned}$$

(1.4.1)

Let us first recall some basic but important information about (n, w) due to the presence of the quadratic degradation term in the first equation of (1.4.1).

Lemma 1.36

The classical solution (n, w, u, P) of (1.4.1) satisfies

(i) $\displaystyle \lim \sup _{t\rightarrow \infty }\Vert n(\cdot ,t)\Vert _{L^1(\varOmega )}\le \frac{|\varOmega |r_+}{\mu };$

(ii)$\displaystyle \int ^{t}_{t_0} \Vert n(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le \frac{r_+}{\mu }\int ^{t}_{t_0}\Vert n(\cdot ,s)\Vert _{L^1(\varOmega )}ds + \frac{1}{\mu }\Vert n(\cdot ,t_0)\Vert _{L^1(\varOmega )}$ for all $t> t_0$;

(iii) $\displaystyle \int ^{t}_{t_0} \int _\varOmega |\nabla w|^2 dxds\le \int _\varOmega w(x,t_0) dx +\int ^{t}_{t_0}\Vert n(\cdot ,s)\Vert _{L^1(\varOmega )}ds$ for all $t>t_0$.

In particular, if $r\le 0$, then

$$\begin{aligned} \Vert n(\cdot ,t)\Vert _{L^1(\varOmega )}\le \displaystyle \frac{|\varOmega |}{\mu (t+\gamma )}~~~\text {for all}~~ t>t_0\end{aligned}$$

(1.4.2)

with $\gamma =\displaystyle \frac{|\varOmega |}{\mu \int _\varOmega n_0(x) dx}$.

Proof

Integrating the first equation in (1.4.1) and using the Cauchy–Schwarz inequality, we get

$$\begin{aligned} \begin{aligned} \displaystyle \frac{d}{dt}\int _\varOmega n=&\displaystyle r\int _\varOmega n- \mu \int _\varOmega n^2 \le \displaystyle r_+\int _\varOmega n - \frac{\mu }{|\varOmega |}(\int _\varOmega n )^2 \end{aligned} \end{aligned}$$

(1.4.3)

which yields (i) readily. By the time integration of (1.4.3) over $(t_0,t)$, we get (ii) immediately. In addition, from the second equation in (1.4.1), $\nabla \cdot u=0$ and $u=0$ on $\partial \varOmega$, it follows that

$$\begin{aligned} \displaystyle \frac{d}{dt}\int _\varOmega w = -\displaystyle \int _\varOmega |\nabla w|^2 +\int _\varOmega n, \end{aligned}$$

(1.4.4)

and thus establishes (iii).

When $r\le 0$, it follows from (1.4.3) that

$$\begin{aligned} \displaystyle \frac{d}{dt}\int _\varOmega n \le \displaystyle - \frac{\mu }{|\varOmega |}(\int _\varOmega n )^2 \end{aligned}$$

(1.4.5)

which then yields (1.4.2) by the time integration.

In order to make use of the spatio-temporal properties provided by Lemma 1.36(ii) to estimate the ultimate bound of $\int _\varOmega |\nabla u|^2$, we shall utilize the following elementary lemma (see Lemma 3.4 of Winkler 2019a):

Lemma 1.37

Let $t_0\ge 0, T\in (t_0,\infty ]$, $a>0$ and $b>0$, and suppose that the nonnegative function $h\in L^1_{loc}(\mathbb {R})$ satisfies $\int ^{t+1}_{t} h(s)ds\le b$ for all $t\in [t_0,T]$. If $y \in C^0([t_0, T ))\cap C^1([t_0, T ))$ has the property that $y'(t) + ay(t) \le h(t)~~\hbox {for all}~~ t\in (t_0, T ),$ then $y(t)\le e^{-a(t-t_0)}y(t_0)+\frac{b}{1-e^{-a}} ~~\hbox {for all }~~t\in [t_0, T ).$

With Lemmas 1.36 and 1.37 at hand, we can employ the standard energy inequality associated with the fluid evolution system in (1.4.1) to derive some boundedness results for u.

Lemma 1.38

For the global classical solution (n, w, u) of (1.4.1), we have

(i) if $r>0$, then

$$\begin{aligned} \displaystyle \lim \sup _{t\rightarrow \infty }\Vert u(\cdot ,t)\Vert ^2_{L^2(\varOmega )}\le \frac{3(1+r) |\varOmega |}{\mu }\frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{r}{\mu }\end{aligned}$$

(1.4.6)

as well as

$$\begin{aligned} \displaystyle \lim \sup _{t\rightarrow \infty }\int ^{t+1}_{t}\Vert \nabla u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le \frac{5(1+r) |\varOmega |}{\mu }\frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{r}{\mu }\end{aligned}$$

(1.4.7)

with Poincaré constant $C_P>0$.

(ii) if $r\le 0$, then

$$\begin{aligned} \int _\varOmega |u(\cdot ,t)|^2\le \Vert u(\cdot ,t_0)\Vert ^2_{L^2(\varOmega )}e^{-\frac{C_p}{2}(t-t_0)}+ \frac{2 |\varOmega |}{\mu ^2} \frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{1}{t_0+\gamma } ~~~~~\hbox {for all}~~t>t_0 \end{aligned}$$

(1.4.8)

as well as

$$\begin{aligned}&\int ^{t+1}_{t}\Vert \nabla u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\nonumber \\ \le&\Vert u(\cdot ,t_0)\Vert ^2_{L^2(\varOmega )}e^{-\frac{C_p}{2}(t-t_0)}+ \frac{4 |\varOmega |}{\mu ^2} \frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{1}{t_0+\gamma } ~~~\hbox {for all}~t>t_0. \end{aligned}$$

(1.4.9)

Proof

(i) According to the Poincaré inequality, one can find some constant $C_p>0$ such that $C_p\int _\varOmega |u|^2\le \int _\varOmega |\nabla u|^2.$ Testing the third equation in (1.4.1) by u and using the Hölder inequality, we obtain

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

due to $u|_{\partial \varOmega } = 0$ and $\nabla \cdot u = 0$.

Writing $h(t)=\frac{2}{C_p}\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\Vert n(\cdot ,t)\Vert _{L^2(\varOmega )}^2$, we see that $y(t):=\int _\varOmega |u(\cdot ,t)|^2$ satisfies

$$\begin{aligned} y'(t) +\frac{C_p}{2} y(t)+ \int _\varOmega |\nabla u(\cdot ,t)|^2 \le h(t)~~~~~\hbox {for all}~~t>0. \end{aligned}$$

(1.4.10)

In view of Lemma 1.36 (i) and (ii), we know that

$$\begin{aligned} \displaystyle \lim \sup _{t\rightarrow \infty }\int ^{t+1}_{t} h(s)ds\le \frac{2}{C_p}\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )} \frac{(1+r) |\varOmega |}{\mu }\frac{r}{\mu }. \end{aligned}$$

(1.4.11)

An application of Lemma 1.37 thus shows that there exists positive $t_0>0$ such that

$$\int _\varOmega |u(\cdot ,t)|^2\le \Vert u(\cdot ,t_0)\Vert ^2_{L^2(\varOmega )}e^{-\frac{C_p}{2}(t-t_0)}+ \frac{3(1+r) |\varOmega |}{\mu }\frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{r}{\mu }~~~~~\hbox {for all}~~t>t_0 $$

and thereby verifies (1.4.6). Thereafter, again thanks to (1.4.11), an integration of (1.4.10) in time yields (1.4.7).

(ii) In view of (1.4.2), we have

$$\begin{aligned} \int ^{t+1}_{t} h(s)ds\le \frac{2}{C_p}\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )} \frac{ |\varOmega |}{\mu ^2} \frac{1}{t+\gamma }, \end{aligned}$$

(1.4.12)

whereupon Lemma 1.37 guarantees that

$$\int _\varOmega |u(\cdot ,t)|^2\le \Vert u(\cdot ,t_0)\Vert ^2_{L^2(\varOmega )}e^{-\frac{C_p}{2}(t-t_0)}+ \frac{2 |\varOmega |}{\mu ^2} \frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{1}{t_0+\gamma } ~~~~~\hbox {for all}~~t>t_0. $$

This precisely warrants (1.4.8), and thereby in turn yields (1.4.9) after integrating (1.4.10) over $(t,t+1)$ and once more employing (1.4.12).

Now by a further testing procedure, we can turn the above information into the estimate of $\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}$, particularly its decay in the case of $r=0$, on the basis of an interpolation argument, which is inspired by an approach illustrated in section 3.2 of Tao and Winkler (2016).

Lemma 1.39

For the global classical solution (n, w, u, P) of (1.4.1), we have

(i) if $r>0$, then there exists $\mu _1:=\mu _1(\varOmega , r)>0$ such that for all $\mu >\mu _1$,

$$\begin{aligned} \displaystyle \lim \sup _{t\rightarrow \infty }\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}\le \frac{1}{17K_1|\varOmega |} \end{aligned}$$

(1.4.13)

(ii) if $r\le 0$, then for any $\mu >0$,

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}=0. \end{aligned}$$

(1.4.14)

Proof

Applying the Helmholtz projector $\mathscr {P}$ to the third equation in (1.4.1), multiplying the resulting identity $u_t + Au = -\mathscr {P}[(u \cdot \nabla )u] +\mathscr {P}[n\nabla \phi ]$ by Au, and using the Gagliardo–Nirenberg inequality, we can find $C_1>0$ such that

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _{\varOmega }|\nabla u|^2+\int _\varOmega |A u|^2\\&=-\int _\varOmega \mathscr {P} [(u \cdot \nabla )u] \cdot A u+\int _\varOmega \mathscr {P}[n \nabla \phi ] \cdot A u \\&\le \frac{1}{2}\int _\varOmega |A u|^2+\int _\varOmega |(u \cdot \nabla )u|^2+\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _\varOmega n^2\\&\le \frac{1}{2}\int _\varOmega |A u|^2+\Vert u\Vert ^2_{L^\infty (\varOmega )} \Vert \nabla u\Vert ^2_{L^2(\varOmega )}+\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _\varOmega n^2\\&\le \frac{1}{2}\int _\varOmega |A u|^2+C_1 \Vert A u\Vert _{L^2(\varOmega )}\Vert u\Vert _{L^2(\varOmega )} \Vert \nabla u\Vert ^2_{L^2(\varOmega )}+ \Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _\varOmega n^2\\&\le \int _\varOmega |A u|^2+\frac{C^2_1}{2} \Vert u\Vert ^2_{L^2(\varOmega )} \Vert \nabla u\Vert ^4_{L^2(\varOmega )}+ \Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int _\varOmega n^2, \end{aligned}$$

which entails $y(t):=\int _\varOmega |\nabla u(\cdot ,t)|^2$ satisfies

$$\begin{aligned} y'(t) \le h_1(t)y(t)+h_2(t)~~~~~\hbox {for all}~~t>0 \end{aligned}$$

(1.4.15)

with $h_1(t){=}C^2_1\Vert u(\cdot ,t)\Vert ^2_{L^2(\varOmega )} \Vert \nabla u(\cdot ,t)\Vert ^2_{L^2(\varOmega )}$ and $h_2(t)=2\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\Vert n (\cdot ,t)\Vert ^2_{L^2(\varOmega )}$.

(i) In order to prepare the integration of (1.4.15), we may use Lemma 1.38 (i) to find some $t_0>0$ such that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert ^2_{L^2(\varOmega )}\le C_2:=\frac{3(1+r) |\varOmega |}{\mu }\frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{r}{\mu }\end{aligned}$$

and $\int ^{t}_{t-1}\Vert \nabla u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le 2 C_2$ for all $t>t_0+1$.

Hence for any $t>t_0+1$, we can find $t_*=t_*(t)\in [t-1,t)$ such that

$$\begin{aligned} \Vert \nabla u(\cdot ,t_*)\Vert ^2_{L^2(\varOmega )}\le 2 C_2, \end{aligned}$$

(1.4.16)

and then integrating (1.4.15) over $(t_*,t)$ yields

$$ y(t)\le y(t_*)e^{\int ^t_{t_*}h_1(\sigma )d\sigma } + \int ^t_{t_*} e^{\int ^t_{s}h_1(\sigma )d\sigma } h_2(s)ds\le (2+C_p)C_2e^{2C_1^2C_2^2} $$

and thereby verifies (1.4.13).

(ii) For any $t_0>1$ and $t>t_0+2$, we use Lemma 1.38 (ii) to pick $t_*=t_*(t)\in [t-1,t)$ fulfilling

$$\begin{aligned} \Vert \nabla u(\cdot ,t_*)\Vert ^2_{L^2(\varOmega )}=&\int ^{t}_{t-1}\Vert \nabla u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\\ \le&\Vert u(\cdot ,t_0)\Vert ^2_{L^2(\varOmega )}e^{-\frac{C_p}{2}(t-1-t_0)}+ \frac{4 |\varOmega |}{\mu ^2} \frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{1}{t_0+\gamma }, \end{aligned}$$

as well as

$$\begin{aligned} \int ^t_{t-1}h_1(\sigma )d\sigma&\le C^2_1\max _{t-1\le s\le t}\Vert u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}\int ^t_{t-1} \Vert \nabla u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\\&\le C^2_1 (\Vert u(\cdot ,t_0)\Vert ^2_{L^2(\varOmega )}e^{-\frac{C_p}{2}(t-1-t_0)}+ \frac{4 |\varOmega |}{\mu ^2} \frac{\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}}{C_p(1-e^{-\frac{C_p}{2}})}\frac{1}{t_0+\gamma })^2. \end{aligned}$$

In addition, by (1.4.12) we also have

$$\begin{aligned} \int ^t_{t-1}h_2(\sigma )d\sigma = 2\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )}\int ^t_{t-1}\Vert n (\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds \le 2\Vert \nabla \phi \Vert ^2_{L^\infty (\varOmega )} \frac{ |\varOmega |}{\mu ^2} \frac{1}{t-1+\gamma }.\end{aligned}$$

Therefore combining the above inequalities, (1.4.15) implies that

$$\begin{aligned} y(t)&\le y(t_*)e^{\int ^t_{t-1}h_1(\sigma )d\sigma } + e^{\int ^t_{t-1}h_1(\sigma )d\sigma } \int ^t_{t-1} h_2(s)ds \end{aligned}$$

and thus (1.4.14) holds readily.

1.4.2 Global Boundedness of Solutions

In this party, we show that the classical solution of problem (1.4.1) is globally bounded in the cases of $r>0$ and $r\le 0$, respectively.

1. The Case $r>0$

In this subsection, we derive the global boundedness of solutions to (1.4.1) whenever $\mu$ is suitably large compared with r. As in Winkler (2016c), the main idea is to examine the behavior of the functional

$$\begin{aligned} \mathscr {F}(n,w):= \int _\varOmega H(n) +\frac{\chi }{2}\displaystyle \int _\varOmega |\nabla w|^2 \end{aligned}$$

(1.4.17)

where $H(s):=s\ln \frac{\mu s}{er}+ \frac{r}{\mu }$, along trajectories of the boundary value problem (1.4.1).

The following elementary property of H(n) will be used in the sequel.

Lemma 1.40

For all nonnegative function $n\in C(\bar{\varOmega })$, $H(n)\ge 0$.

Proof

It is easy to verify that $H(\frac{r}{\mu })=0$, $H'(\frac{r}{\mu })=0$ and $H''(s)=\frac{1}{s}\ge 0$, which implies $H(n)\ge 0$ for all $n\ge 0$.

Now we can describe the evolution of $\mathscr {F}(n,w)$ along the trajectories of (1.4.1) by the standard testing procedure.

Lemma 1.41

Let $\varOmega \subset \mathbb {R}^2$ be a smooth bounded domain and (n, w, u) be the global classical solution of (1.4.1) with $r>0, \mu >0$. Then whenever $\mu >\mu _2(\varOmega ,\chi ,r):=\max \{\mu _1,\frac{K_1(36+16\chi )|\varOmega |}{ \chi }r\}$, there exists $t_*>0$ such that

$$\begin{aligned} \frac{d}{dt}{\mathscr {F}(n,w)}\le 0 ~~\hbox {for all}~~ t\ge t_*. \end{aligned}$$

(1.4.18)

Proof

Multiplying the first equation in (1.4.1) by $H'(n)$ and integrating by parts, we get

$$\begin{aligned} \begin{aligned} \displaystyle \frac{d}{dt}\int _\varOmega H(n)=&\displaystyle \int _\varOmega H'(n)(\triangle n+\chi \nabla \cdot ( n\nabla w)+rn- \mu n^2-u\cdot \nabla n)\\ =&-\displaystyle \int _\varOmega H''(n) ( |\nabla n|^2 +\chi n \nabla n\cdot \nabla w)+ \displaystyle \int _\varOmega H'(n)(rn- \mu n^2)\\ =&-\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}-\chi \displaystyle \int _\varOmega \nabla n\cdot \nabla w + \displaystyle \displaystyle \int _\varOmega (\ln n-\ln \frac{r}{\mu }) (rn- \mu n^2)\\ \le&-\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}-\chi \displaystyle \int _\varOmega \nabla n\cdot \nabla w \end{aligned} \end{aligned}$$

(1.4.19)

due to $(\ln n-\ln \frac{r}{\mu }) (rn- \mu n^2)\le 0$, $\nabla \cdot u=0$ and $u=0$ on $\partial \varOmega$.

On the other hand, testing the second equation in (1.4.1) by $-\triangle w$, using $\nabla \cdot u=0$ and $u=0$ on $\partial \varOmega$ again, we can obtain

$$\begin{aligned}&\displaystyle \frac{1}{2}\frac{d}{dt} \int _\varOmega |\nabla w|^2+ \int _\varOmega |\triangle w|^2 \nonumber \\ =&\displaystyle \int _\varOmega |\nabla w|^2 \triangle w+ \displaystyle \int _\varOmega \nabla n\cdot \nabla w+\int _\varOmega (u\cdot \nabla w) \triangle w \nonumber \\ \le&\displaystyle \frac{1}{2} \int _\varOmega |\triangle w|^2+ \displaystyle \frac{1}{2} \int _\varOmega |\nabla w|^4 +\displaystyle \int _\varOmega \nabla n\cdot \nabla w+\int _\varOmega (u\cdot \nabla w) \triangle w \nonumber \\ =&\displaystyle \frac{1}{2} \int _\varOmega |\triangle w|^2+ \displaystyle \frac{1}{2} \int _\varOmega |\nabla w|^4 +\displaystyle \int _\varOmega \nabla n\cdot \nabla w-\int _\varOmega \nabla w\cdot (\nabla u\cdot \nabla w) - \int _\varOmega u(D^2w \cdot \nabla w) \nonumber \\ =&\displaystyle \frac{1}{2} \int _\varOmega |\triangle w|^2+ \displaystyle \frac{1}{2} \int _\varOmega |\nabla w|^4 +\displaystyle \int _\varOmega \nabla n\cdot \nabla w-\int _\varOmega \nabla w\cdot (\nabla u\cdot \nabla w) - \displaystyle \frac{1}{2}\int _\varOmega u \cdot \nabla |\nabla w|^2 \nonumber \\ =&\displaystyle \frac{1}{2} \int _\varOmega |\triangle w|^2+ \displaystyle \frac{1}{2} \int _\varOmega |\nabla w|^4 +\displaystyle \int _\varOmega \nabla n\cdot \nabla w-\int _\varOmega \nabla w\cdot (\nabla u\cdot \nabla w) . \nonumber \end{aligned}$$

Furthermore, by Lemma 1.9 (i) and the Cauchy–Schwarz inequality, we get

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{2}\frac{d}{dt} \int _\varOmega |\nabla w|^2+ \displaystyle \frac{1}{2} \int _\varOmega |\triangle w|^2\\ \le&\displaystyle \frac{K_1}{2}\Vert \nabla w\Vert ^2_{L^2(\varOmega )} \int _\varOmega |\triangle w|^2 +\displaystyle \int _\varOmega \nabla n\cdot \nabla w+\displaystyle \int _\varOmega |\nabla u||\nabla w|^2\\ \le&( \displaystyle \frac{K_1}{2}\Vert \nabla w\Vert ^2_{L^2(\varOmega )} +K_1|\varOmega |^{\frac{1}{2}}\Vert \nabla u\Vert _{L^2(\varOmega )}) \int _\varOmega |\triangle w|^2 +\displaystyle \int _\varOmega \nabla n\cdot \nabla w \end{aligned} \end{aligned}$$

and thus

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{2}\displaystyle \frac{d}{dt} \int _\varOmega |\nabla w|^2+ \displaystyle \frac{1}{2}(1-K_1\Vert \nabla w\Vert ^2_{L^2(\varOmega )} -2 K_1|\varOmega |^{\frac{1}{2}}\Vert \nabla u\Vert _{L^2(\varOmega )}) \int _\varOmega |\triangle w|^2\\ \le&\displaystyle \int _\varOmega \nabla n\cdot \nabla w. \end{aligned} \end{aligned}$$

(1.4.20)

Since $2{\mathscr {F}(n,w)}\ge \chi \Vert \nabla w\Vert ^2_{L^2(\varOmega )}$ due to $H(n)\ge 0$, combining (1.4.20) with (1.4.19) yields

$$\begin{aligned} \frac{d}{dt}{\mathscr {F}(n,w)} +\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}+ (\displaystyle \frac{\chi }{2}- K_1{\mathscr {F}(n,w)}-2\chi K_1|\varOmega |^{\frac{1}{2}}\Vert \nabla u\Vert _{L^2(\varOmega )} ) \int _\varOmega |\triangle w|^2\le 0 \end{aligned}$$

(1.4.21)

for $t>0$.

On the other hand, when $\mu >\mu _1$, it follows from (1.4.13) that it is possible to pick some $t_0>0$ such that $16 K_1|\varOmega |^{\frac{1}{2}}\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}<1 ~~~\hbox {for all }~ t>t_0,$ and thereby

$$\begin{aligned} \frac{d}{dt}{\mathscr {F}(n,w)} +\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}+ (\displaystyle \frac{3\chi }{8}- K_1\mathscr {F}(n,w)) \int _\varOmega |\triangle w|^2 \le 0 ~~\hbox {for }~ t>t_0. \end{aligned}$$

(1.4.22)

In what follows, we shall show that there exists $t_*>t_0$ such that $4 K_1\mathscr {F}(n,w)(t_*)<\chi$.

Firstly by Lemma 1.36 (i), there exists $t_1>t_0$ such that for all $t>t_1$

$$\begin{aligned} \Vert n(\cdot ,t)\Vert _{L^1(\varOmega )}\le \frac{3|\varOmega |r}{2\mu }, \end{aligned}$$

(1.4.23)

which along with Lemma 1.36 (iii) yields

$$\begin{aligned} \begin{aligned} \displaystyle \int ^{t_2}_{t_1} \int _\varOmega |\nabla w|^2 \le&\displaystyle \int _\varOmega w(\cdot ,t_1) +\int ^{t_2}_{t_1}\Vert n(\cdot ,s)\Vert _{L^1(\varOmega )}ds\\ \le&\displaystyle \int _\varOmega w_0(x) +\int ^{t_1}_{0}\Vert n(\cdot ,s)\Vert _{L^1(\varOmega )}ds+\frac{3|\varOmega |r}{2\mu }(t_2-t_1). \end{aligned} \end{aligned}$$

Similarly invoking Lemma 1.36 (i) and (ii), we find that

$$\displaystyle \int ^{t_2}_{t_1} \Vert n(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le \frac{3|\varOmega |}{2}(\frac{r}{\mu })^2 (t_2-t_1) + \frac{1}{\mu }\Vert n(\cdot ,t_1)\Vert _{L^1(\varOmega )}. $$

Hence there exists $t^*>t_1$ suitably large such that whenever $t_2\ge t^*$,

$$\begin{aligned} \displaystyle \int ^{t_2}_{t_1} \int _\varOmega |\nabla w|^2 \le \frac{2|\varOmega |r}{\mu }(t_2-t_1) \end{aligned}$$

(1.4.24)

and

$$\begin{aligned} \displaystyle \int ^{t_2}_{t_1} \Vert n(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds \le 2|\varOmega | (\frac{r}{\mu })^2 (t_2-t_1). \end{aligned}$$

(1.4.25)

Let

$$\mathscr {S}_1:=\{t\in [t_1,t_2]| \int _\varOmega |\nabla w(\cdot ,t)|^2\ge \frac{8|\varOmega |r}{\mu }\} $$

and

$$\mathscr {S}_2:=\{t\in [t_1,t_2]| \Vert n(\cdot ,t)\Vert ^2_{L^2(\varOmega )}\ge 8|\varOmega | (\frac{r}{\mu })^2 \}. $$

Then

$$\begin{aligned} |\mathscr {S}_1|\le \frac{|t_2-t_1|}{4}, ~~~|\mathscr {S}_2|\le \frac{|t_2-t_1|}{4}. \end{aligned}$$

(1.4.26)

In order to estimate the size of $\mathscr {S}_1$ and $\mathscr {S}_2$, we recall (1.4.24) to get

$$\begin{aligned} \frac{8|\varOmega |r}{\mu }|\mathscr {S}_1|\le \displaystyle \int ^{t_2}_{t_1} \int _\varOmega |\nabla w|^2 \le \frac{2|\varOmega |r}{\mu }(t_2-t_1) \end{aligned}$$

and thus $|\mathscr {S}_1|\le \frac{|t_2-t_1|}{4}$ is valid. Similarly, one can verify that $|\mathscr {S}_2|\le \frac{|t_2-t_1|}{4}$.

As (1.4.26) warrants that $|(t_1,t_2)\setminus (\mathscr {S}_1\cup \mathscr {S}_2)| \ge \frac{|t_2-t_1|}{2},$ one can conclude that there exists $t_*\in (t_1,t_2)$ such that

$$\begin{aligned} \displaystyle \Vert n(\cdot ,t_*)\Vert ^2_{L^2(\varOmega )}< 8|\varOmega | (\frac{r}{\mu })^2 \end{aligned}$$

(1.4.27)

and

$$\begin{aligned} \int _\varOmega |\nabla w(\cdot ,t_*)|^2<\frac{8|\varOmega |r}{\mu }. \end{aligned}$$

(1.4.28)

Applying $\xi \ln \frac{\xi }{\sigma }\le \eta \xi ^2+ \ln \frac{1}{\eta \sigma }\cdot \xi$ for all $\xi>0, \eta>0, \sigma >0$ with $\eta =\frac{\mu }{r}$ (see Lemma 5.5 of Winkler 2016c) and (1.4.27), we then arrive at

$$ \displaystyle \int _\varOmega H(n)(\cdot ,t_*) \le \displaystyle \frac{\mu }{ r}\int _\varOmega n^2(\cdot ,t_*) -\int _\varOmega n(\cdot ,t_*) +\frac{r}{\mu }|\varOmega | \le \displaystyle \frac{9 |\varOmega |r}{\mu }. $$

Thereupon from (1.4.28) and the definition of $\mathscr {F}(n,w)$, it follows that $\mathscr {F}(n,w)(t_*)< (9+ 4\chi )|\varOmega |\frac{r}{\mu },$ which entails that $4 K_1\mathscr {F}(n,w)(t_*){<}\chi$ provided $\mu > \frac{K_1(36+16\chi )|\varOmega |r}{ \chi }$.

As an immediate consequence of (1.4.22), we have

$$\begin{aligned} \frac{d}{dt}{\mathscr {F}(n,w)} +\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}+ \displaystyle \frac{\chi }{8} \int _\varOmega |\triangle w|^2\le 0 ~~\hbox {for all}~ t>t_* \end{aligned}$$

(1.4.29)

when $\mu >\mu _2(\varOmega ,\chi ,r)$, and thus end the proof of this lemma.

Additionally from (1.4.29), one can also conclude that

Corollary 1.1

Under the conditions of Lemma 1.41, we have

$$\begin{aligned} \mathscr {F}(n,w)(t) +\displaystyle \displaystyle \int ^\infty _{t_*}\int _\varOmega \frac{ |\nabla n|^2}{n}+ \displaystyle \frac{\chi }{8} \int ^\infty _{t_*}\int _\varOmega |\triangle w|^2\le (9+ 4 \chi )|\varOmega |\frac{r}{\mu }~~~\hbox {for all}~ t>t_*. \end{aligned}$$

(1.4.30)

Next by a further testing procedure, we can turn the above information into the uniform-in-time boundedness of $\Vert n(\cdot ,t)\Vert _{L^2(\varOmega )}$ and $\Vert \nabla w(\cdot ,t)\Vert _{L^4(\varOmega )}$ if $\mu$ is appropriately large compared with r, which will serve as the foundation for the proof of global boundedness of $\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ and $\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}$.

Lemma 1.42

If $\mu >\mu _0(\chi , \varOmega , r):=\max \{\mu _2(\chi , \varOmega , r), \frac{208 K_2|\varOmega |r}{\chi ^2}\}$, then there exists $C>0$ such that

$$\begin{aligned} \displaystyle \Vert n(\cdot ,t)\Vert _{L^2(\varOmega )}+\Vert \nabla w(\cdot ,t)\Vert _{L^4(\varOmega )}\le C~~\hbox {for all}~~ t\ge t_*. \end{aligned}$$

(1.4.31)

Proof

Since $\mu >\mu _2(\chi , \varOmega , r)$, it follows from (1.4.30) that

$$ \int _{\varOmega }|\nabla w|^2\le \frac{r}{\mu }(\frac{18}{\chi }+\frac{8}{\chi ^2} )|\varOmega |~~\hbox { for all}~t>t_* $$

and moreover due to $\frac{r}{\mu }< \frac{\chi ^2}{208 K_2|\varOmega |}$,

$$\begin{aligned} K_2\int _{\varOmega } |\nabla w|^2\le \frac{1}{8}~~\hbox { for all}~t>t_*. \end{aligned}$$

(1.4.32)

Multiplying the first equation in (1.4.1) by n and integrating the result over $\varOmega$, we get

$$\begin{aligned} \begin{aligned} \displaystyle \frac{1}{2} \frac{d}{dt}\int _\varOmega n^2=&-\displaystyle \int _\varOmega |\nabla n|^2- \chi \int _\varOmega n \nabla n \nabla w +r \int _\varOmega n^2- \mu \int _\varOmega n^3\\ \le&-\displaystyle \frac{1}{2}\displaystyle \int _\varOmega |\nabla n|^2+ \frac{1}{2} \int _\varOmega n^2 | \nabla w|^2 +r \int _\varOmega n^2- \mu \int _\varOmega n^3. \end{aligned} \end{aligned}$$

(1.4.33)

On the other hand, by the second equation in (1.4.1) and the identity $\nabla w\cdot \nabla \varDelta w=\frac{1}{2} \varDelta |\nabla w|^2- |D^2 w|^2$, we obtain

$$\begin{aligned}&\frac{d}{dt}\int _{\varOmega }|\nabla w|^4\nonumber \\&\qquad =\, 2\int _{\varOmega }|\nabla w|^2 \triangle |\nabla w|^2- 4 \int _{\varOmega }|\nabla w|^2 |D^2 w|^2- 4 \int _{\varOmega }|\nabla w|^2 \nabla w \cdot \nabla |\nabla w |^2\nonumber \\&\qquad \quad +\, 4 \int _{\varOmega }|\nabla w|^2 \nabla n \cdot \nabla w - 4 \int _{\varOmega }|\nabla w|^2 \nabla w\cdot \nabla ( u \cdot \nabla w)\nonumber \\&\qquad = - 2\int _{\varOmega }|\nabla |\nabla w|^2|^2- 4 \int _{\varOmega }|\nabla w|^2 |D^2 w|^2 \nonumber \\&\qquad \quad -4 \int _{\varOmega }|\nabla w|^2 \nabla w \cdot \nabla |\nabla w |^2-4 \int _{\varOmega }n |\nabla w|^2 \triangle w \nonumber \\&\qquad \quad -4 \int _{\varOmega }n \nabla |\nabla w|^2\cdot \nabla w + 2 \int _{\partial \varOmega } |\nabla w|^2 \frac{ \partial |\nabla w|^2}{\partial \nu }- 4 \int _{\varOmega }|\nabla w|^2 \nabla w\cdot (\nabla u \cdot \nabla w)\nonumber \\ \end{aligned}$$

(1.4.34)

due to $\nabla \cdot u=0$ and $u=0$ on $\partial \varOmega$.

According to $\frac{ \partial |\nabla w|^2}{\partial \nu } \le C_1|\nabla w|^2~~\hbox {on} ~\partial \varOmega ~\hbox {for some}~ C_1>0$ and

$$ \Vert |\nabla w|^2\Vert _{L^2(\partial \varOmega )}\le \eta \Vert \nabla |\nabla w|^2\Vert _{L^2(\varOmega )}+C_2(\eta )\Vert |\nabla w|^2\Vert _{L^1(\varOmega )}~\hbox {for any}~ \eta \in (0,\frac{5}{4}) $$

(see Lemma 4.2 of Mizoguchi and Souplet (2014) and Remark 52.9 in Quittner and Souplet 2007), one can conclude that

$$\begin{aligned} 2 \displaystyle \int _{\partial \varOmega } |\nabla w|^2 \frac{ \partial |\nabla w|^2}{\partial \nu }\le \frac{1}{4}\int _{\varOmega }|\nabla |\nabla w|^2|^2 +C_3(\int _{\varOmega }|\nabla w|^2)^2 \end{aligned}$$

(1.4.35)

for some $C_3>0$.

For the other integrals on the right side of (1.4.34), we use the Young inequality to estimate

$$\begin{aligned} -4 \int _{\varOmega }|\nabla w|^2 \nabla w \cdot \nabla |\nabla w |^2 \le \displaystyle \frac{1}{3}\int _{\varOmega }|\nabla |\nabla w|^2|^2+12 \int _{\varOmega }|\nabla w|^6 \end{aligned}$$

(1.4.36)

$$\begin{aligned} -4\displaystyle \int _{\varOmega }n \nabla |\nabla w|^2 \cdot \nabla w \le \displaystyle \frac{1}{3}\int _{\varOmega }|\nabla |\nabla w|^2|^2+12 \int _{\varOmega }n^2|\nabla w|^2 \end{aligned}$$

(1.4.37)

as well as

$$\begin{aligned} \begin{aligned} -4\displaystyle \int _{\varOmega }n |\nabla w|^2 \triangle w&\le \displaystyle \frac{1}{6} \int _{\varOmega } |\nabla w|^2 |\triangle w|^2+24\int _{\varOmega }n^2|\nabla w|^2 \\&\le \displaystyle \frac{1}{3} \int _{\varOmega } |\nabla w|^2 |D^2 w|^2+24\int _{\varOmega }n^2|\nabla w|^2 \end{aligned} \end{aligned}$$

(1.4.38)

due to $|\triangle w|^2\le 2|D^2 w|^2$ on $\varOmega$.

Substituting (1.4.35)–(1.4.38) into (1.4.34), we readily get

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }|\nabla w|^4+ \frac{13}{12}\int _{\varOmega }|\nabla |\nabla w|^2|^2+ \frac{11}{3}\int _{\varOmega }|\nabla w|^2 |D^2 w|^2 \\ \le&\displaystyle 12\int _{\varOmega } |\nabla w|^6+36 \int _{\varOmega }n^2|\nabla w|^2+C_3(\int _{\varOmega }|\nabla w|^2)^2+4\int _{\varOmega } |\nabla w|^4 |\nabla u| \end{aligned} \end{aligned}$$

and thus

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }|\nabla w|^4+ 2\int _{\varOmega }|\nabla |\nabla w|^2|^2\\ \le&\displaystyle 12\int _{\varOmega } |\nabla w|^6+36 \int _{\varOmega }n^2|\nabla w|^2+C_3(\int _{\varOmega }|\nabla w|^2)^2+4\int _{\varOmega } |\nabla w|^4 |\nabla u| \end{aligned} \end{aligned}$$

(1.4.39)

due to the fact $|\nabla |\nabla w|^2|^2\le 4 |\nabla w|^2 |D^2 w|^2 ~\hbox {on}~\varOmega$.

Therefore combining (1.4.33) with (1.4.39) leads to

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }(n^2+ |\nabla w|^4)+ 2\int _{\varOmega }|\nabla |\nabla w|^2|^2+ \displaystyle \int _\varOmega |\nabla n|^2 \\ \le&\displaystyle 12\int _{\varOmega } |\nabla w|^6+37 \int _{\varOmega }n^2|\nabla w|^2+C_3(\int _{\varOmega }|\nabla w|^2)^2 \\ {}&+2r \int _\varOmega n^2- 2\mu \int _\varOmega n^3 +4\int _{\varOmega } |\nabla w|^4 |\nabla u|\\ \le&\displaystyle 13\int _{\varOmega } |\nabla w|^6+37^2 \int _{\varOmega }n^3+C_3(\int _{\varOmega }|\nabla w|^2)^2 \\&+2r \int _\varOmega n^2- 2\mu \int _\varOmega n^3+4\int _{\varOmega } |\nabla w|^4 |\nabla u|. \end{aligned} \end{aligned}$$

(1.4.40)

Furthermore by Lemma 1.9 (ii), we get $\Vert \varphi \Vert ^3_{L^3}\le K_2\Vert \nabla \varphi \Vert ^2_{L^2}\Vert \varphi \Vert _{L^1}+ C_4\Vert \varphi \Vert ^3_{L^1}$ and thus

$$ \int _{\varOmega } |\nabla w|^6\le K_2(\int _{\varOmega }|\nabla |\nabla w|^2|^2)\left( \int _{\varOmega } |\nabla w|^2\right) +C_4(\int _{\varOmega } |\nabla w|^2)^3. $$

Upon inserting this into (1.4.40) and (1.4.30), we obtain

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega } (n^2+ |\nabla w|^4)+ (2-13K_2\int _{\varOmega } |\nabla w|^2)\int _{\varOmega }|\nabla |\nabla w|^2|^2 \\&+\displaystyle \int _\varOmega |\nabla n|^2+ \int _{\varOmega }(n^2+ |\nabla w|^4) \\&\le \displaystyle 37^2 \int _{\varOmega }n^3+(2r+1) \int _\varOmega n^2- 2\mu \int _\varOmega n^3+ \displaystyle \int _{\varOmega }|\nabla w|^4+4\int _{\varOmega } |\nabla w|^4 |\nabla u|+C_5, \end{aligned} \end{aligned}$$

which, along with

$$ \int _{\varOmega }|\nabla w|^4\le \frac{1}{7}\int _{\varOmega }|\nabla |\nabla w|^2|^2 +C_6 $$

and

$$ 4 \displaystyle \int _{\varOmega } |\nabla w|^4 |\nabla u| \le 4\Vert |\nabla w|^2\Vert ^2_{L^6(\varOmega )}\Vert \nabla u\Vert _{L^{\frac{3}{2}}(\varOmega )} \le \displaystyle \frac{13}{56} \int _{\varOmega }|\nabla |\nabla w|^2|^2 +C_7 $$

by the Gagliardo–Nirenberg inequality and (1.4.13), implies that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }(n^2+ |\nabla w|^4)+ (\frac{13}{8}-13K_2\int _{\varOmega } |\nabla w|^2)\int _{\varOmega }|\nabla |\nabla w|^2|^2 \\&+\displaystyle \int _\varOmega |\nabla n|^2+ \int _{\varOmega } (n^2+ |\nabla w|^4) \\&\le \displaystyle 37^2 \int _{\varOmega }n^3+(2r+1) \int _\varOmega n^2- 2\mu \int _\varOmega n^3+C_8. \end{aligned} \end{aligned}$$

(1.4.41)

On the other hand, according to an extended variant (Biler et al. 1994), (1.4.23) and (1.4.30), one can infer that

$$ 37^2 \displaystyle \int _{\varOmega }n^3\le C_9\displaystyle \left( \int _\varOmega |\nabla n|^2\right) \left( \int _\varOmega n |\ln n|\right) +C_9 (\int _\varOmega n)^3+C_9 \le \displaystyle \frac{1}{2} \int _\varOmega |\nabla n|^2+C_{10}. $$

Hence from (1.4.41) it follows that there exists $C_{11}>0$ such that for all $t>t_*$

$$\begin{aligned} \displaystyle \frac{d}{dt}\int _{\varOmega }(n^2+ |\nabla w|^4)+ \displaystyle \int _{\varOmega } (n^2+ |\nabla w|^4) +(\frac{13}{8}-13K_2\int _{\varOmega } |\nabla w|^2)\int _{\varOmega }|\nabla |\nabla w|^2|^2 \le C_{11}, \end{aligned}$$

(1.4.42)

which, along with (1.4.32), entails that

$$ \displaystyle \frac{d}{dt}\int _{\varOmega }(n^2+ |\nabla w|^4)+ \displaystyle \int _{\varOmega } (n^2+ |\nabla w|^4) \le C_{11} $$

for all $t>t_*$ and thereby (1.4.31) is valid.

We are now ready to prove Theorem 1.2 in the case of $r>0$.

Proof of Theorem 1.2 in the case of $r>0$.  From the above lemmas, it follows that there exists $C>0$ such that

$$ \displaystyle \Vert n(\cdot ,t)\Vert _{L^2(\varOmega )}+\Vert \nabla w(\cdot ,t)\Vert _{L^4(\varOmega )}+\Vert \nabla u(\cdot ,t)\Vert _{L^2(\varOmega )}\le C~~ $$

whenever $\mu >\mu _0(\chi , \varOmega , r):=\max \{\mu _2(\chi , \varOmega , r), \frac{208 K_2|\varOmega |r}{\chi ^2}\}$. So, by the argument in, e.g., Lemma 4.4 of Black (2018), we can readily prove that $\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )},\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ and $\Vert A^\alpha u(\cdot ,t)\Vert _{L^2(\varOmega )}$ with some $\alpha \in (\frac{1}{2},1)$ are globally bounded; we refer the reader to the proof of Lemma 4.4 in Black (2018), Lemmas 3.12 and 3.11 in Tao and Winkler (2016) for the details.

Based on the global boundedness of solutions, we are able to derive the convergence result claimed in Theorem 1.2, namely,

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert n(\cdot ,t)-\frac{r}{\mu }\Vert _{L^\infty (\varOmega )}=0, \end{aligned}$$

(1.4.43)

$$\begin{aligned} ~~\displaystyle \lim _{t\rightarrow \infty }\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}=0, \end{aligned}$$

(1.4.44)

$$\begin{aligned} ~~\displaystyle \lim _{t\rightarrow \infty }\Vert u (\cdot ,t)\Vert _{L^\infty (\varOmega )}=0 \end{aligned}$$

(1.4.45)

as well as

$$\begin{aligned} ~~\displaystyle \lim _{t\rightarrow \infty }\displaystyle \inf _{x\in \varOmega } w(x,t) =\infty . \end{aligned}$$

(1.4.46)

In fact, due to

$$ \int ^{\infty }_{t_*}\int _\varOmega \frac{ |\nabla n|^2}{n}+ \int ^{\infty }_{t_*}\int _\varOmega |\triangle w|^2\le C $$

established in (1.4.30), we can show (1.4.44), (1.4.45) and

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert n(\cdot ,t)-\overline{n}(t)\Vert _{L^\infty (\varOmega )}=0 \end{aligned}$$

(1.4.47)

with $\overline{n}(t)=\frac{1}{|\varOmega |}\int _\varOmega n(\cdot , t)$ by the arguments in Proposition 4.15 of Black (2018), where we have used

$$ \displaystyle \int ^{\infty }_{t_*}\Vert n(\cdot ,t)-\overline{n}(t)\Vert ^2_{L^2(\varOmega )} \le C\displaystyle \int ^{\infty }_{t_*}\Vert \nabla n\Vert ^2_{L^1(\varOmega )} \le C\displaystyle \int ^{\infty }_{t_*}(\int _\varOmega \frac{ |\nabla n|^2}{n}\int _\varOmega n) $$

and the regularity of n. Therefore it suffices to show that

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }|\overline{n}(t)-\frac{r}{\mu }|=0. \end{aligned}$$

(1.4.48)

To this end, we adapt the idea of Liţcanu and Morales-Rodrigo (2010b) and give the details of the proof for the convenience of readers.

Integrating the first equation in (1.4.1) on the spatial variable over $\varOmega$, we obtain

$$ \overline{n}_t= r\overline{n} - \frac{\mu }{|\varOmega |} \int _\varOmega n^2 =r \overline{n} -\mu \overline{n}^2 - \frac{\mu }{|\varOmega |}\int _\varOmega (n-\overline{n})^2. $$

Putting $a(t):= \frac{\mu }{|\varOmega |}\int _\varOmega (n(\cdot ,t)-\overline{n})^2$, the above equation then becomes

$$\begin{aligned} \overline{n}_t= \mu \overline{n}(\frac{r}{\mu }- \overline{n}) - a(t). \end{aligned}$$

(1.4.49)

Thereupon multiplying (1.4.49) by $\overline{n}-\frac{r}{\mu }$, we get

$$\begin{aligned} \frac{d}{dt}(\overline{n}-\frac{r}{\mu })^2+ 2\mu \overline{n}(\overline{n}-\frac{r}{\mu })^2 = -2 a(t)(\overline{n}-\frac{r}{\mu }) \end{aligned}$$

and then

$$\begin{aligned} 2\mu \int ^\infty _{1}\overline{n}(\overline{n}-\frac{r}{\mu })^2\le (\overline{n}(1)-\frac{r}{\mu })^2+ 2 \sup _{t\ge 1}|\overline{n}(t)-\frac{r}{\mu }| \int ^\infty _{1}a(t). \end{aligned}$$

(1.4.50)

In addition, invoking the Poincaré–Wirtinger inequality

$$\displaystyle \int _\varOmega |\varphi -\frac{1}{|\varOmega |}\int _\varOmega \varphi (y)dy|^2 \le C_p\int _\varOmega |\varphi | \int _\varOmega \frac{|\nabla \varphi |^2}{|\varphi |} ~~\hbox { for all~} \varphi \in W^{1,2}(\varOmega )$$

for some $C_p > 0$, one can find

$$\begin{aligned} \displaystyle \int ^{\infty }_1 a(s)ds \le C_p\displaystyle \sup _{t\ge 1}\Vert n(t)\Vert _{L^1(\varOmega )} \displaystyle \int ^{\infty }_1 \int _\varOmega \frac{|\nabla n(s)|^2}{n(s)} ds\le C \end{aligned}$$

(1.4.51)

due to (1.4.30) and Lemma 1.38 (i). Hence combining (1.4.51) with (1.4.50) yields

$$\begin{aligned} \int ^\infty _{1}\overline{n}(\overline{n}-\frac{r}{\mu })^2\le C. \end{aligned}$$

(1.4.52)

On the other hand, $\frac{d}{dt}\overline{n}(\overline{n}-\frac{r}{\mu })^2=\overline{n}_t((\overline{n}-\frac{r}{\mu })^2+2\overline{n}(\overline{n}-\frac{r}{\mu })),$ which along with $|\overline{n}_t|\le r\overline{n} + \frac{\mu }{|\varOmega |} \int _\varOmega n^2\le C$ implies that

$$\begin{aligned} \left| \frac{d}{dt}\overline{n}(\overline{n}-\frac{r}{\mu })^2\right| \le C. \end{aligned}$$

(1.4.53)

Therefore by Lemma 6.3 of Liţcanu and Morales-Rodrigo (2010b), (1.4.53) and (1.4.52) show that

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty } \overline{n}(t)(\overline{n}(t)-\frac{r}{\mu })^2=0. \end{aligned}$$

(1.4.54)

From (1.4.47), it follows that there exists $t_1>t_*$ such that $\Vert n(\cdot ,t)-\overline{n}(t)\Vert _{L^\infty (\varOmega )}\le \frac{r}{2\mu }$ for all $t>t_1$, and thus

$$\begin{aligned} \begin{aligned} \overline{n}_t=&r \displaystyle \overline{n} -\mu \overline{n}^2 - \displaystyle \frac{\mu }{|\varOmega |}\int _\varOmega n(n-\overline{n})\\ \ge&\mu \overline{n}(\displaystyle \frac{r}{\mu }-\overline{n}-\sup _{t>t_1}\Vert n(\cdot ,t)-\overline{n}(t)\Vert _{L^\infty (\varOmega )})\\ \ge&\mu \overline{n}(\displaystyle \frac{r}{2\mu }-\overline{n}). \end{aligned} \end{aligned}$$

(1.4.55)

On the other hand, noticing that the solution y(t) of the ODE

$$ y'(t)=\mu \overline{y}(\displaystyle \frac{r}{2\mu }-\overline{y}), ~~y(t_1)>0 $$

satisfies $\displaystyle \lim _{t\rightarrow \infty } y(t)=\frac{r}{2\mu },$ by the comparison principle, (1.4.55) implies that there exists $t_2>t_1$ such that for all $t\ge t_2$, $\overline{n}(t)\ge \displaystyle \frac{r}{4\mu }.$ This together with (1.4.54) yields (1.4.48).

Finally, in view of (1.4.43), one can find $t_3>1$ such that $n(x,t)\ge \frac{r}{2\mu }$ for all $x \in \varOmega$ and $t\ge t_3$, and thereby w(x, t) satisfies $w_t\ge \triangle w- |\nabla w|^2 +\frac{r}{2\mu }-u\cdot \nabla w$ for $t\ge t_3$. Hence if y(t) denotes the solution of ODE: $y'(t)=\frac{r}{2\mu },~~y(t_3)=\displaystyle \min _{x\in \varOmega }w(\cdot ,t_3)$, then

$$\begin{aligned} w(x,t)\ge \frac{r}{2\mu }(t-t_3)\end{aligned}$$

(1.4.56)

by means of a straightforward parabolic comparison which warrants that (1.4.46) holds and thereby completes the proof.

2. The Case $r\le 0$

In this subsection, we show the global boundedness of solutions to (1.1.5), (1.1.8), (1.1.9) in the case $r\le 0,\mu >0$. As mentioned in the introduction, due to the structure of (1.1.5) with $r\le 0,\mu >0$, it is difficult to find a decreasing energy functional compared with the situation when $r>0,\mu >0$ considered in the previous subsection or when $r=\mu =0$ considered in Winkler (2016c). Indeed, the energy-type functional $\mathscr {F}(n,w)$ in (3.1) of Winkler (2016c) decreases along a solution in $\varOmega \times (t_0,\infty )$ if $\mathscr {F}(n(\cdot ,t_0),w(\cdot ,t_0))$ is suitably small, namely

$$\frac{d}{dt}{\mathscr {F}(n,w)}\le 0 ~~\hbox {for all}~~ t\ge t_0. $$

The main idea underlying our approach is to make use of the quadratic degradation in the first equation of (1.1.5) which should enforce some suitable regularity properties. More precisely, on the basis of (1.4.2), we can show that the quantity of form

$$\begin{aligned} \mathscr {F}(n,w):= \int _\varOmega n(\ln n+a) dx+\frac{\chi }{2}\displaystyle \int _\varOmega |\nabla w|^2 dx, \end{aligned}$$

(1.4.57)

with parameter $a>0$ determined below (see (1.4.64)), satisfies a certain of differential inequality. Although unlike the case of $r>0$ in which it enjoys the monotonicity property, $\mathscr {F}(n,w)$ also provides us the global boundedness of $\int _\varOmega n|\ln n| dx$ and $\int _\varOmega |\nabla w|^2 dx$. This is encapsulated in the following lemma.

Lemma 1.43

Let $\varOmega \subset \mathbb {R}^2$ be a smooth bounded domain and (n, w, u) be the global classical solution (1.4.1) with $r\le 0, \mu >0$. Then there exists $t_*>0$ such that for all $t> t_*$

$$\begin{aligned} \int _\varOmega |\nabla w(\cdot ,t)|^2\le \frac{1}{4 K_1} \end{aligned}$$

(1.4.58)

with $K_1$ given in Lemma 1.9 as well as

$$\begin{aligned} \int _\varOmega n|\ln n| \le C \end{aligned}$$

(1.4.59)

for some $C>0$.

Proof

We test the first equation in (1.4.1) against $\ln n+a+1$, and integrate by parts to see that

$$\begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega n(\ln n+a)\nonumber \\ \le&-\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}-\chi \displaystyle \int _\varOmega \nabla n\cdot \nabla w + \displaystyle \displaystyle \int _\varOmega (n(r- \mu n)-u\cdot \nabla n) (\ln n+a+1) \nonumber \\ \le&-\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}-\chi \displaystyle \int _\varOmega \nabla n\cdot \nabla w + \displaystyle \displaystyle \int _\varOmega n(r- \mu n) (\ln n+a) \end{aligned}$$

(1.4.60)

due to $r\le 0$ and $\nabla \cdot u=0$.

On the other hand, recalling (1.4.20) and (1.4.14), it is possible to fix $t_0>0$ such that for all $t\ge t_0$, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{2}\frac{d}{dt} \int _\varOmega |\nabla w|^2+ \displaystyle \frac{1}{4}\int _\varOmega |\triangle w|^2+ \displaystyle \frac{1}{4}(\frac{3}{4}-2K_1\Vert \nabla w\Vert ^2_{L^2(\varOmega )} ) \int _\varOmega |\triangle w|^2 \\ \le&\displaystyle \int _\varOmega \nabla u\cdot \nabla w. \end{aligned} \end{aligned}$$

(1.4.61)

From Lemma 1.9 (i), there exists a constant $K_3>0$ such that

$$\begin{aligned} 8 K_3\Vert \nabla w \Vert ^2_{L^2(\varOmega )}\le \Vert \varDelta w\Vert ^2_{L^2(\varOmega )}. \end{aligned}$$

(1.4.62)

Hence combining (1.4.61) with (1.4.60), we get

$$\begin{aligned}&\displaystyle \frac{d}{dt}{\mathscr {F}(n,w)} +\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}+\displaystyle \frac{\chi }{4}\int _\varOmega |\triangle w|^2+ K_3 \int _\varOmega n(\ln n+a)\nonumber \\&+\frac{\chi }{4} (\frac{3}{4}-2 K_1\Vert \nabla w\Vert ^2_{L^2(\varOmega )} ) \int _\varOmega |\triangle w|^2\nonumber \\ \le&\displaystyle \int _\varOmega n(K_3- \mu n)(\ln n+a)+r\displaystyle \int _\varOmega n(\ln n+a)~~\hbox {for }~t\ge t_0. \end{aligned}$$

(1.4.63)

Now for any fixed $\varepsilon <\min \{\frac{\chi }{24K_1}, \frac{\chi }{42K_2}\}$, we pick $a>1$ sufficiently large such that

$$\begin{aligned} e^{-a}< \frac{K_3}{\mu }, ~~~(1-r)|\varOmega |\displaystyle \max _{0<n\le e^{-a}} |n\ln n|< \varepsilon \displaystyle \min \{K_3,1\}, \end{aligned}$$

(1.4.64)

due to $\displaystyle \lim _{n \rightarrow 0} n\ln n=0$ and $n\ln n<0$ for all $n\in (0,1)$, and thereby fix $t_1>\max \{1,t_0\}$ fulfilling

$$\begin{aligned} ~~\displaystyle \frac{a|\varOmega |}{\mu (t_1+\gamma )}<\frac{\varepsilon }{4}, \displaystyle \frac{|\varOmega |}{\mu ^2(t_1+\gamma )}<\frac{\varepsilon }{16} \end{aligned}$$

(1.4.65)

as well as

$$\begin{aligned} \displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )|\varOmega |}{\mu (t_1+\gamma )}+\frac{2\chi ( \frac{|\varOmega |}{\mu }+\Vert n_0\Vert _{L^1(\varOmega )}+\Vert w_0\Vert _{L^1(\varOmega )})}{t_1}< \displaystyle \frac{\varepsilon }{4}. \end{aligned}$$

(1.4.66)

Let $t_2=t_1+t_1^2$,

$$\mathscr {S}_1\triangleq \{t\in [t_1,t_2]| \int _\varOmega |\nabla w(\cdot ,t)|^2\ge \frac{\varepsilon }{2\chi } \} $$

and

$$\mathscr {S}_2\triangleq \{t\in [t_1,t_2]| \Vert n(\cdot ,t)\Vert ^2_{L^2(\varOmega )}\ge \frac{\varepsilon }{4} \}. $$

Then

$$\begin{aligned} |\mathscr {S}_1|\le \frac{|t_2-t_1|}{4}, ~~~|\mathscr {S}_2|\le \frac{|t_2-t_1|}{4}. \end{aligned}$$

(1.4.67)

By Lemma 1.36(iii), (1.4.2) and the second equation in (1.4.1), we obtain that

$$\begin{aligned} \displaystyle \int ^{t_2}_{t_1} \int _\varOmega |\nabla w|^2 \le&\int ^{t_2}_{t_1} \int _\varOmega n + \int _\varOmega w(\cdot ,t_1)\nonumber \\ =&\int ^{t_2}_{t_1} \int _\varOmega n +\int _\varOmega w_0+\int ^{t_1}_{0} \int _\varOmega n\nonumber \\ \le&\displaystyle \frac{|\varOmega |}{\mu (t_1+\gamma )}(t_2-t_1)+ \int _\varOmega w_0+t_1\int _\varOmega n_0\nonumber . \end{aligned}$$

Furthermore, by (1.4.65) and (1.4.66)

$$\begin{aligned} \displaystyle \int ^{t_2}_{t_1} \int _\varOmega |\nabla w|^2 dxds\le&\displaystyle (\frac{|\varOmega |}{\mu (t_1+\gamma )}+ \frac{t_1\Vert n_0\Vert _{L^1(\varOmega )}+\Vert w_0\Vert _{L^1(\varOmega )}}{t_2-t_1})(t_2-t_1)\nonumber \\ \le&\frac{\frac{|\varOmega |}{\mu }+\Vert n_0\Vert _{L^1(\varOmega )}+\Vert w_0\Vert _{L^1(\varOmega )}}{t_1}(t_2-t_1)\\ <&\frac{\varepsilon }{8\chi }(t_2-t_1). \end{aligned}$$

On the other hand, by the definition of $\mathscr {S}_1$, we see that $\displaystyle \frac{\varepsilon }{2\chi }|\mathscr {S}_1|\le \int ^{t_2}_{t_1} \int _\varOmega |\nabla w|^2$ and thereby $|\mathscr {S}_1|\le \frac{|t_2-t_1|}{4}$.

In addition, by (1.4.2) and (1.4.65), we get

$$\displaystyle \int ^{t_2}_{t_1} \int _\varOmega n^2 \le \frac{1}{\mu }\int _\varOmega n(\cdot ,t_1)\le \displaystyle \frac{|\varOmega |}{\mu ^2(t_1+\gamma )}<\frac{\varepsilon }{16},$$

which implies that $|\mathscr {S}_2|\le \frac{|t_2-t_1|}{4}$.

Therefore from (1.4.67), it follows that $|(t_1,t_2)\setminus (\mathscr {S}_1\cup \mathscr {S}_2)| \ge \frac{|t_2-t_1|}{2},$ and thereby there exists $t_*\in (t_1,t_2)$ such that

$$\begin{aligned} \displaystyle \Vert n(\cdot ,t_*)\Vert ^2_{L^2(\varOmega )}< \frac{\varepsilon }{4} \end{aligned}$$

(1.4.68)

and

$$\begin{aligned} \int _\varOmega |\nabla w(\cdot ,t_*)|^2<\frac{\varepsilon }{2\chi }<\frac{1}{6K_1}. \end{aligned}$$

(1.4.69)

By (1.4.69), we can see that the set

$$ \mathbf {S}\triangleq \{ t\in ( t_*,\infty )| ~K_1\int _\varOmega |\nabla w(\cdot ,s)|^2<\frac{1}{4} \quad \hbox {for all}\, s\in ( t_*,t)\} $$

is not empty and hence $T_S=\sup {\mathbf {S}}$ is a well-defined element of $(t_*,\infty ]$. In fact, we claim that $T_S=\infty$. To this end, supposing on the contrary that $T_S<\infty$, we then have $K_1\int _\varOmega |\nabla w(\cdot ,t)|^2<\frac{1}{4}$ for all $t\in [t_*,T_S)$, but

$$\begin{aligned} K_1\int _\varOmega |\nabla w(\cdot ,T_S)|^2=\frac{1}{4}. \end{aligned}$$

(1.4.70)

Hence from (1.4.63) and (1.4.62), it follows that for all $t\in [t_*,T_S)$,

$$\begin{aligned}&\displaystyle \frac{d}{dt}{\mathscr {F}(n,w)} +\displaystyle \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n}+\displaystyle \frac{\chi }{4}\int _\varOmega |\triangle w|^2+K_3 \int _\varOmega n(\ln n+a)+\frac{K_3\chi }{2} \int _\varOmega |\nabla w|^2\nonumber \\ \le&\displaystyle \int _\varOmega n(K_3- \mu n)(\ln n+a)+ r \int _\varOmega n(\ln n+a)\nonumber \\ \le&\displaystyle \int _{e^{-a}<n\le \frac{K_3}{\mu }}n(K_3- \mu n)(\ln n+a)+ \displaystyle r\int _{0<n\le e^{-a}}n(\ln n+a)\nonumber \\ \le&K_3\displaystyle \int _{e^{-a}<n\le \frac{K_3}{\mu }}n(\ln n+a)+ \displaystyle r\int _{0<n\le e^{-a}}n\ln n \\ \le&aK_3\displaystyle \int _\varOmega n+K_3 \displaystyle \int _{e^{-a}<n\le \frac{K_3}{\mu }}n\ln n -r|\varOmega |\displaystyle \max _{0<n\le e^{-a}} |n\ln n| \nonumber \\ \le&K_3 (a+(\ln \frac{K_3}{\mu })_+ )\displaystyle \int _\varOmega n + \varepsilon K_3\nonumber \\ \le&\displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )K_3|\varOmega |}{\mu (t_1+\gamma )}+ \varepsilon K_3,\nonumber \end{aligned}$$

(1.4.71)

where we have made use of $t_*\ge t_1$, the decay estimate (1.4.2) and (1.4.65), and thus

$$\begin{aligned}&\mathscr {F}(n,w)(T_s)+\displaystyle \int ^{T_s}_{t_*} \displaystyle e^{- K_3(T_s-\sigma )}(\int _\varOmega \frac{ |\nabla n|^2}{n}(\cdot ,\sigma )+\displaystyle \frac{\chi }{4}\int _\varOmega |\triangle w(\cdot ,\sigma )|^2)d\sigma \\ \le&\mathscr {F}(n,w)(t_*)+\displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )|\varOmega |}{\mu (t_1+\gamma )}+ \varepsilon , \end{aligned}$$

which implies that

$$\begin{aligned} \displaystyle \frac{\chi }{2}\int _\varOmega |\nabla w(\cdot ,T_S)|^2 \le&\mathscr {F}(n,w)(t_*)+\displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )|\varOmega |}{\mu (t_1+\gamma )}-\int _\varOmega n(\ln n+a)(\cdot ,T_S)+ \varepsilon \nonumber \\ \le&\int _\varOmega n(\ln n+a) (\cdot ,t_*)+\frac{\chi }{2}\displaystyle \int _\varOmega |\nabla w|^2(\cdot ,t_*)+ \varepsilon \nonumber \\&+\displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )|\varOmega |}{\mu (t_1+\gamma )}-\int _\varOmega n(\ln n+a)(\cdot ,T_S)\nonumber \\ \le&\int _\varOmega (n^2+an) (\cdot ,t_*)+\frac{\chi }{2}\displaystyle \int _\varOmega |\nabla w|^2(\cdot ,t_*)+ \varepsilon \\&+\displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )|\varOmega |}{\mu (t_1+\gamma )}-\int _\varOmega n(\ln n+a)(\cdot ,T_S),\nonumber \end{aligned}$$

(1.4.72)

due to $n\ge \ln n$ for all $n>0$.

In addition, by (1.4.65), we see that

$$\begin{aligned} \int _\varOmega n(\ln n+a)(\cdot ,T_S)\ge&\int _{0<n\le e^{-a}} n(\ln n+a)(\cdot ,T_S)\\ \ge&\int _{0<n\le e^{-a}} n\ln n (\cdot ,T_S)\nonumber \\ \ge&-|\varOmega |\displaystyle \max _{0<n\le e^{-a}} |n\ln n|\nonumber \\ \ge&-\varepsilon \nonumber . \end{aligned}$$

(1.4.73)

Upon inserting (1.4.73) into (1.4.72), we see that

$$\begin{aligned} \displaystyle \frac{\chi }{2}\int _\varOmega |\nabla w(\cdot ,T_S)|^2 \le&\int _\varOmega (n^2+an) (\cdot ,t_*)+\frac{\chi }{2}\displaystyle \int _\varOmega |\nabla w|^2(\cdot ,t_*){c1.2-3.58}\\&+\displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )|\varOmega |}{\mu (t_1+\gamma )}+2\varepsilon ,\nonumber \end{aligned}$$

(1.4.74)

which along with (1.4.68), (1.4.69), (1.4.2) and (1.4.65), establishes that

$$\begin{aligned} \displaystyle \frac{\chi }{2}\int _\varOmega |\nabla w(\cdot ,T_S)|^2 \le&\displaystyle \frac{5\varepsilon }{2}+ a\int _\varOmega n (\cdot ,t_*)+\displaystyle \frac{(a+(\ln \frac{K_3}{\mu })_+ )|\varOmega |}{\mu (t_1+\gamma )}\nonumber \\ <&3\varepsilon \\ \le&\frac{\chi }{8K_1}. \nonumber \end{aligned}$$

(1.4.75)

This contradicts (1.4.70) and thereby $T_S=\infty$, which means that the differential inequality (1.4.71) is actually valid for all $t>t_*$.

Now revisiting the proof of (1.4.75), upon integration in time over $(t_*,t)$, we have $\frac{\chi }{2}\int _\varOmega |\nabla w(\cdot ,t)|^2 \le 3\varepsilon$ for all $t>t_*$ which implies that (1.4.58) is valid by the choice of $\varepsilon$, as well as

$$\begin{aligned} \int _\varOmega n \ln n(\cdot ,t) \le C_1~~~\hbox {for all}~~t>t_* \end{aligned}$$

(1.4.76)

for some $C_1>0$.

Since $\xi \ln \xi \ge -\frac{1}{e}$ for all $\xi >0$,

$$ \int _\varOmega n |\ln n|(\cdot ,t) = \int _\varOmega n \ln n(\cdot ,t)-2 \int _{0<n<1} n \ln n(\cdot ,t) \le \int _\varOmega n \ln n(\cdot ,t)+\frac{2|\varOmega |}{e}, $$

which along with (1.4.76) readily implies that (1.4.59) is actually valid with $C=C_1+\frac{2|\varOmega |}{e}$.

Furthermore, from (1.4.71), one can also conclude that

Corollary 1.2

Under the conditions of Lemma 1.43, we have

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\int ^{t+1}_{t}\int _\varOmega ( \frac{ |\nabla n|^2}{n}+ |\triangle w|^2) =0, \quad \displaystyle \lim _{t\rightarrow \infty }\int _\varOmega |\nabla w(\cdot ,t)|^2=0. \end{aligned}$$

(1.4.77)

Proof

On the basis of the decay estimate (1.4.2) and revisiting the argument in the proof of Lemma 1.43, one can conclude that for any $\varepsilon \in (0,\min \{\frac{\chi }{24K_1}, \frac{\chi }{42K_2}\})$, there exists $t_\varepsilon >1$ such that

$$ \int _\varOmega |\nabla w(\cdot ,t)|^2+\displaystyle \int ^{t}_{t_\varepsilon } \displaystyle e^{- K_3(t-\sigma )} (\int _\varOmega \frac{ |\nabla n|^2}{n}(\cdot ,\sigma )+\displaystyle \frac{\chi }{8}\int _\varOmega |\triangle w(\cdot ,\sigma )|^2)d\sigma \le \varepsilon $$

for all $t>t_\varepsilon$. Furthermore, it follows from the above inequality that

$$ \displaystyle \int ^{t}_{t-1} \displaystyle (\int _\varOmega \frac{ |\nabla n|^2}{n}(\cdot ,\sigma )+ \displaystyle \frac{\chi }{8}\int _\varOmega |\triangle w(\cdot ,\sigma )|^2)d\sigma \le \varepsilon e^{ K_3} $$

for any $t>t_\varepsilon +1$, which implies that (1.4.77) is indeed valid.

At this point, we can prove Theorem 1.2 in the case of $r\le 0$.

Proof of Theorem 1.2 in the case $r\le 0$.  We can repeat the argument in the proof of Theorem 1.2 in the case $r>0$. In fact, in view of (1.4.58) and (1.4.59), (1.4.31) is also valid for $r\le 0,\mu >0$, and thereby the global boundedness of solutions can be proven. In addition, similar to the case of $r>0$, we can show

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}=0, \end{aligned}$$

(1.4.78)

$$\begin{aligned} ~~\displaystyle \lim _{t\rightarrow \infty }\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}=0 \end{aligned}$$

(1.4.79)

as well as

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}=0. \end{aligned}$$

(1.4.80)

For the sake of completeness we shall only recount the main steps and refer to the mentioned sources for more details. Invoking standard parabolic regularity theory (see the proofs of Lemma 4.5 and Lemma 4.9 of Winkler 2016c for details), one can see that there exist $\theta \in (0,1)$ and $\alpha \in (\frac{1}{2},1)$ and $C_1>0$ such that for all $t>1$

$$\begin{aligned} \Vert n\Vert _{C^{\theta ,\frac{\theta }{2}}(\overline{\varOmega }\times [t,t+1])}+\Vert \nabla w(\cdot ,t)\Vert _{C^{\theta }(\overline{\varOmega })}+ \Vert A^\alpha u(\cdot ,t)\Vert _{L^2(\varOmega )} \le C_1. \end{aligned}$$

(1.4.81)

If (1.4.78) were false, then there would be $C_2>0$, $(t_k)_{k\in \mathbb {N}}$ and $(x_k)_{k\in \mathbb {N}}\subseteq \varOmega$ such that $t_k\rightarrow \infty$ as $k\rightarrow \infty$, and $n(x_k,t_k)>C_2$ for all $k\in \mathbb {N}$, which, along with the uniform continuity of n in $\overline{\varOmega }\times [t,t+1]$ as shown by (1.4.81), entails that one can find $r>0$ such that $B(x_k, r)\subseteq \varOmega$ for all $k\in \mathbb {N}$ and $n(x,t_k)>\frac{C_2}{2}~~\hbox {for all}~~x\in B(x_k, r).$ This shows

$$ \int _{\varOmega }n(\cdot ,t_k)\ge \int _{B(x_k, r)}n(\cdot ,t_k)\ge \frac{C_2}{2} \pi r^2 $$

which contradicts (1.4.2) and thus proves (1.4.78). Similarly, on the basis of (1.4.77) and (1.4.81), (1.4.79) can be proved. Finally, (1.4.80) results from (1.4.14), (1.4.81) and a simple interpolation, and thereby completes the proof.

1.4.3 Asymptotic Profile of Solutions

It is observed that in the case $r<0$, solutions to (1.1.5), (1.1.8), (1.1.9) enjoy the exponential decay property due to the exponential decay of $\Vert n(\cdot ,t)\Vert _{L^1(\varOmega )}$. Therefore, we pay our attention to the asymptotic profile of (1.1.5), (1.1.8), (1.1.9) in the cases $r>0$ and $r=0$, namely, we will give the proofs of Theorems 1.3 and 1.4 respectively.

1. The Case $r>0$

Making use of the convergence properties of $(n,\frac{|\nabla c|}{c})$ asserted in Theorem 1.2, we apply $L^p-L^q$ estimates for the Neumann heat semigroup $(e^{t\varDelta })_{t>0}$ to show $(n,c,u)\rightarrow (\frac{r}{\mu },0,0)$ in $L^\infty (\varOmega )$ and $\frac{|\nabla c|}{c} \rightarrow 0$ in $L^p(\varOmega )$ at some exponential rate as $t\rightarrow \infty$, respectively, whenever $\mu$ is suitably large compared with r. To this end, we first make an observation which will be used in the proof of the subsequent lemma:

Lemma 1.44

For any fixed $\alpha \in (0,\min \{\lambda _1, r\})$, there exists $\varepsilon _1>0$ such that

$$\begin{aligned} 8 \mu \varepsilon _1<r-\alpha \end{aligned}$$

(1.4.82)

as well as

$$\begin{aligned} 4 c_1I \varepsilon _2<1,~~8\chi c_4 I\varepsilon _2<1 \end{aligned}$$

(1.4.83)

where $c_i>0$ (i=1,4) is given in Lemma 1.1, $I=\int _0^\infty (1+\sigma ^{-\frac{2}{3}}+\sigma ^{-\frac{1}{2}})e^{-(\lambda _1-\alpha ) \sigma }d\sigma$ and $\varepsilon _2= 4 c_1|\varOmega |^{\frac{1}{6}} I \varepsilon _1$.

Lemma 1.45

Let (n, w, u) be the global bounded solution of (1.4.1). For fixed $\alpha \in (0,\min \{\lambda _1, r\})$ and $\mu > 32 \chi c_4c_1|\varOmega |^{\frac{1}{6}}I^2 r$, one can find constants $C_i>0$ ($i=1,2,3$) and $\beta <\alpha$ such that

$$\begin{aligned} \Vert n(\cdot ,t)-\frac{r}{\mu }\Vert _{L^\infty (\varOmega )}&\le C_1e^{-\alpha t}, \end{aligned}$$

(1.4.84)

$$\begin{aligned} \Vert \nabla w(\cdot ,t)\Vert _{L^6(\varOmega )}&\le C_2e^{-\alpha t} \end{aligned}$$

(1.4.85)

as well as

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}&\le C_3 e^{-\beta t} \end{aligned}$$

(1.4.86)

for all $t\ge 1$.

Proof

Let $\tilde{N}(x,t)=n(x,t)-\frac{r}{\mu }$, $\varepsilon _1>0$ and $\varepsilon _2>0$ be given by Lemma 1.44. Then from (1.4.43), (1.4.44) and (1.4.45), there exists $t_0>1$ suitably large such that for $t\ge t_0$

$$\begin{aligned} \Vert \tilde{N}(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \frac{\varepsilon _1}{8},~~~(c_2+1)\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )} \le \frac{\varepsilon _2}{8} \end{aligned}$$

(1.4.87)

and

$$\begin{aligned} 8c_1\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\int _0^\infty (1+\sigma ^{-\frac{1}{2}})e^{-(\lambda _1-\alpha ) \sigma }d\sigma \le 1. \end{aligned}$$

(1.4.88)

Now we consider

$$\begin{aligned} T\!\triangleq \!\sup \!\left\{ \!\widetilde{T}\!\in \!(t_0,\infty )\!\left| \begin{aligned} \!\Vert \tilde{N}(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \varepsilon _1 e^{-\alpha (t-t_0)} \quad \hbox {for all }\, t\in [t_0,\widetilde{T}), \\ \!\Vert \nabla w(\cdot ,t)\Vert _{L^6(\varOmega )}\le \varepsilon _2 e^{-\alpha (t-t_0)} \quad \hbox {for all }\, t\in [t_0,\widetilde{T}). \end{aligned}\right. \!\right\} \end{aligned}$$

(1.4.89)

By (1.4.87), T is well-defined. In what follows, we shall demonstrate that $T=\infty$.

To this end, we first invoke the variation-of-constants representation of w:

$$\begin{aligned} \begin{aligned} w(\cdot ,t)=&e^{(t-t_0) \varDelta }w(\cdot ,t_0) -\displaystyle \int _{t_0}^t e^{(t-s)\varDelta }|\nabla w(\cdot ,s)|^2 ds+ \int _{t_0}^t e^{(t-s)\varDelta } \tilde{N} (\cdot ,s) ds\\&\displaystyle -\int _{t_0}^t e^{(t-s)\varDelta }(u\cdot \nabla w)(\cdot ,s) ds+\frac{r}{\mu }(t-t_0), \end{aligned} \end{aligned}$$

(1.4.90)

and use Lemma 1.1(i), (ii) to estimate

$$\begin{aligned} \begin{aligned}&\Vert \nabla w(\cdot ,t)\Vert _{L^6(\varOmega )} \\ \le&\displaystyle \Vert \nabla e^{(t-t_0) \varDelta }w(\cdot ,t_0)\Vert _{L^6(\varOmega )} +\displaystyle \int _{t_0}^t \Vert \nabla e^{(t-s)\varDelta }|\nabla w(\cdot ,s)|^2\Vert _{L^6(\varOmega )} ds \\&+\displaystyle \int _{t_0}^t \Vert \nabla e^{(t-s)\varDelta }N(\cdot ,s)\Vert _{L^6(\varOmega )} ds +\displaystyle \int _{t_0}^t \Vert \nabla e^{(t-s)\varDelta }(u\cdot \nabla w)(\cdot ,s)\Vert _{L^6(\varOmega )} ds \\ \le&2 c_2e^{-\lambda _1(t-t_0)}\Vert \nabla w(\cdot ,t_0)\Vert _{L^6(\varOmega )} \\&+c_1\displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{2}{3}})e^{-\lambda _1(t-s)}\Vert \nabla w(\cdot ,s)\Vert ^2_{L^6(\varOmega )}ds \\&+c_1|\varOmega |^{\frac{1}{6}}\displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{1}{2}})e^{-\lambda _1(t-s)}\Vert \tilde{N}(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}ds \\&+c_1\displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{1}{2}})e^{-\lambda _1(t-s)}\Vert u(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}\Vert \nabla w(\cdot ,s)\Vert _{L^6(\varOmega )}ds \\ :=&I_1+I_2+I_3 \end{aligned} \end{aligned}$$

(1.4.91)

for all $t_0<t<T$.

Now we estimate the terms $I_i$ $(i=1,2,3)$, respectively. Firstly, from (1.4.87), we have $I_1\le \frac{\varepsilon _2}{4}e^{-\lambda _1(t-t_0)}$. By the definition of T and (1.4.83), we can see that

$$\begin{aligned} \begin{aligned} I_2\le&c_1\varepsilon ^2_2\displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{2}{3}})e^{-\lambda _1(t-s)}e^{-2\alpha (s-t_0)}ds\\ \le&c_1\varepsilon ^2_2\displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{2}{3}})e^{-\lambda _1(t-s)}e^{-\alpha (s-t_0)}ds \\ \le&c_1\varepsilon ^2_2\displaystyle \int _{0}^\infty (1+\sigma ^{-\frac{2}{3}})e^{-(\lambda _1-\alpha )\sigma }d \sigma \cdot e^{-\alpha (t-t_0)}\\ \le&\displaystyle \frac{\varepsilon _2}{4}e^{-\alpha (t-t_0)}. \end{aligned} \end{aligned}$$

By the definition of T, (1.4.88) and $\varepsilon _2= 4 c_1|\varOmega |^{\frac{1}{6}} I \varepsilon _1$, we also have

$$\begin{aligned} \begin{aligned} I_3\le&(c_1|\varOmega |^{\frac{1}{6}}\varepsilon _1+c_1\displaystyle \sup _{t\ge t_0}\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )} \varepsilon _2) \displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{1}{2}})e^{-\lambda _1(t-s)}e^{-\alpha (s-t_0)}ds\\ =&(c_1|\varOmega |^{\frac{1}{6}}\varepsilon _1+c_1\displaystyle \sup _{t\ge t_0}\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )} \varepsilon _2) \displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{1}{2}})e^{-(\lambda _1-\alpha )(t-s)}e^{-\alpha (t-t_0)}ds \\ \le&(c_1|\varOmega |^{\frac{1}{6}}\varepsilon _1+c_1\displaystyle \sup _{t\ge t_0}\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )} \varepsilon _2) \displaystyle \int _{0}^\infty (1+\sigma ^{-\frac{1}{2}})e^{-(\lambda _1-\alpha )\sigma }d \sigma \cdot e^{-\alpha (t-t_0)}\\ \le&\displaystyle \frac{3\varepsilon _2}{8}e^{-\alpha (t-t_0)}. \end{aligned} \end{aligned}$$

Substituting these estimates into (1.4.91), we get

$$\begin{aligned} \Vert \nabla w(\cdot ,t)\Vert _{L^6(\varOmega )}\le \frac{7\varepsilon _2}{8} e^{-\alpha (t-t_0)}< \varepsilon _2 e^{-\alpha (t-t_0)} \quad \hbox {for all }\, t\in [t_0,T). \end{aligned}$$

(1.4.92)

On the other hand, since $\tilde{N}_t=\triangle \tilde{N}+\chi \nabla \cdot ( n\nabla w)-r\tilde{N}- \mu \tilde{N}^2- u\cdot \nabla \tilde{N},$ the variation-of-constants representation of $\tilde{N}$ yields

$$ \begin{aligned} \tilde{N}(\cdot ,t)=&\displaystyle e^{(t-t_0) (\varDelta -r) }\tilde{N}(\cdot ,t_0) +\chi \int _{t_0}^t e^{(t-s)(\varDelta -r)}\nabla \cdot ( n\nabla w)(\cdot ,s)ds \\&- \mu \int _{t_0}^t e^{(t-s)(\varDelta -r)}\tilde{N}^2(\cdot ,s) ds -\displaystyle \int _{t_0}^t e^{(t-s)(\varDelta -r)}( u\cdot \nabla \tilde{N})(\cdot ,s)ds. \end{aligned} $$

Then by $\nabla \cdot u=0$ we can see that

$$\begin{aligned} \begin{aligned}&\Vert \tilde{N}(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\\ \le&\Vert \displaystyle e^{(t-t_0) (\varDelta -r) }\tilde{N}(\cdot ,t_0)\Vert _{L^{\infty }(\varOmega )} +\displaystyle \mu \int _{t_0}^t \Vert e^{(t-s)(\varDelta -r)}\tilde{N}^2(\cdot ,s) \Vert _{L^{\infty }(\varOmega )}ds \\&+\displaystyle \int _{t_0}^t \Vert e^{(t-s)(\varDelta -r)}\nabla \cdot ( u \tilde{N})(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}ds \\&+\displaystyle \chi \int _{t_0}^t \Vert e^{(t-s)(\varDelta -r)}\nabla \cdot ( n\nabla w)(\cdot ,s)\Vert _{L^{\infty }(\varOmega )}ds \\ :=&J_1+J_2+J_3+J_4.\\ \end{aligned} \end{aligned}$$

Here the maximum principle together with (1.4.87) ensures that

$$ J_1\le e^{-r(t-t_0)} \Vert \tilde{N}(\cdot ,t_0)\Vert _{L^\infty (\varOmega )}\le \frac{\varepsilon _1}{8} e^{-\alpha (t-t_0)}. $$

By the definition of T and comparison principle, we infer that

$$\begin{aligned} J_2\le&\displaystyle \mu \int _{t_0}^te^{-r(t-s)} \Vert e^{(t-s)\varDelta } \tilde{N}^2 (\cdot ,s)\Vert _{L^{\infty }(\varOmega )}ds\\ \le&\displaystyle \mu \int _{t_0}^te^{-r(t-s)} \Vert \tilde{N} (\cdot ,s)\Vert ^2 _{L^{\infty }(\varOmega )}ds\\ \le&\mu \varepsilon ^2_1\displaystyle \int _{t_0}^t e^{-r(t-s)} e^{-2\alpha (s-t_0)} ds\\ \le&\mu \varepsilon ^2_1 \displaystyle \int _{t_0}^te^{-(r-\alpha )(t-s)}ds\cdot e^{-\alpha (t-t_0)} \\ \le&\displaystyle \frac{\mu \varepsilon _1^2}{r-\alpha }e^{-\alpha (t-t_0)}\\ \le&\displaystyle \frac{\varepsilon _1}{8} e^{-\alpha (t-t_0)} \end{aligned}$$

due to (1.4.82) and $\alpha < r$. Similarly by (1.4.88), we have

$$\begin{aligned} \begin{aligned} J_3\le&c_1 \displaystyle \sup _{t\ge t_0}\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )} \displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{1}{2}})e^{-(\lambda _1+r)(t-s)} \Vert \tilde{N}(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds \\ \le&c_1 \displaystyle \sup _{t\ge t_0}\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\varepsilon _1 \displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{1}{2}})e^{-(\lambda _1+r)(t-s)} e^{-\alpha (s-t_0)} ds \\ \le&c_1 \displaystyle \sup _{t\ge t_0}\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\varepsilon _1 \int _0^\infty (1+\sigma ^{-\frac{1}{2}})e^{-(\lambda _1-\alpha ) \sigma }d\sigma \cdot e^{-\alpha (t-t_0)}\\ \le&\displaystyle \frac{\varepsilon _1}{8}e^{-\alpha (t-t_0)}. \end{aligned} \end{aligned}$$

As for the term $J_4$, we recall (1.4.83), (1.4.89) and apply Lemma 1.1 (iv) to get

$$ \begin{aligned} J_4\le&\chi c_4 \displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{2}{3}})e^{-(\lambda _1+r)(t-s)} \Vert (n\nabla w)(\cdot ,s)\Vert _{L^6(\varOmega )}ds \\ \le&\chi c_4\varepsilon _2 \displaystyle \int _{t_0}^t(1+(t-s)^{-\frac{2}{3}})e^{-(\lambda _1+r)(t-s)}(\frac{r}{\mu }+ \varepsilon _1 e^{-\alpha (s-t_0)}) e^{-\alpha (s-t_0)} ds \\ \le&\chi c_4\varepsilon _2 (\displaystyle \frac{r}{\mu }+\varepsilon _1)\displaystyle \int _{0}^\infty (1+\sigma ^{-\frac{2}{3}})e^{-(\lambda _1+r-\alpha )\sigma }d \sigma \cdot e^{-\alpha (t-t_0)}\\ \le&\displaystyle \frac{\varepsilon _1}{8}e^{-\alpha (t-t_0)}+\chi c_4 \displaystyle \frac{r}{\mu }I \varepsilon _2 e^{-\alpha (t-t_0)} \\ =&\displaystyle \frac{\varepsilon _1}{8} e^{-\alpha (t-t_0)}+ 4 \chi c_4c_1|\varOmega |^{\frac{1}{6}}I^2\varepsilon _1 \frac{r}{\mu }e^{-\alpha (t-t_0)}\\ \le&\displaystyle \frac{\varepsilon _1}{4} e^{-\alpha (t-t_0)} \end{aligned} $$

due to $\mu > 32 \chi c_4c_1|\varOmega |^{\frac{1}{6}}I^2 r$. Hence combining above inequalities, we arrive at

$$ \Vert \tilde{N}(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \frac{5\varepsilon _1}{8} e^{-\alpha (t-t_0)} \quad \hbox {for all }\, t\in [t_0,T). $$

This along with (1.4.92) readily shows that T cannot be finite. In combination with the decay property (1.4.84), a straightforward interpolation argument can be employed to prove (1.4.86).

Proof of Theorem 1.3.   According to (1.4.56) and $w= -\ln (\frac{c}{\Vert c_0\Vert _{L^\infty (\varOmega )}})$, we have $c(x,t)\le \Vert c_0\Vert _{L^\infty (\varOmega )} e^{-\frac{r}{2\mu }(t-t_3)}$ for all $t\ge t_3$. On the other hand, if $\mu _*(\chi , \varOmega , r) := \max \{\mu _0, 32 \chi c_4c_1|\varOmega |^{\frac{1}{6}}I^2r\}$, then as an immediate consequence of Theorem 1.3 and Lemma 1.45, $n(\cdot ,t) \rightarrow \frac{r}{\mu }$ and $\frac{|\nabla c|}{c}(\cdot ,t)\rightarrow 0$ in $L^\infty (\varOmega )$ and $L^6(\varOmega )$, respectively, at an exponential rate when $\mu >\mu _*(\chi , \varOmega , r)$. Moreover, with the help of the uniform boundedness of $\Vert \frac{|\nabla c|}{c}(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ with respect to $t>0$, one can show that $\frac{|\nabla c|}{c}(\cdot ,t)\rightarrow 0$ in $L^p(\varOmega )$ for any $p>1$ exponentially by the interpolation argument. The proof of this theorem is thus complete.

2. The Case $r=0$

The proof of Theorem 1.4 proceeds on an alternative reasoning. To this end, making use of the decay information on $|\nabla w|$ in $L^\infty (\varOmega )$ in (1.4.77) and the quadratic degradation in the $n-$equation, we first turn the decay property of $\Vert n(\cdot ,t)\Vert _{L^1(\varOmega )}$ from (1.4.2) into an upper bound estimate of $\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}$.

Lemma 1.46

Let (n, w, u) be the global bounded solution of (1.4.1) obtained in Theorem 1.2 with $r=0, \mu >0$. Then one can find constant $C>0$ such that

$$\begin{aligned} \Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \frac{C}{t+1}~~\hbox {for all}~~t>0. \end{aligned}$$

(1.4.93)

Proof

According to the known smoothing properties of the Neumann heat semigroup $(e^{ \tau \varDelta })_{t>0}$ on $\varOmega \subset \mathbb {R}^n$ (see Winkler 2010), one can pick $c_1>0$ and $c_2>0$ such that for all $0<\tau \le 1$,

$$\begin{aligned} \Vert e^{ \tau \varDelta }\varphi \Vert _{L^\infty (\varOmega )}\le C_1\tau ^{-\frac{n}{2}}\Vert \varphi \Vert _{L^1(\varOmega )}~~\hbox {for all}~~ \varphi \in L^1(\varOmega ) \end{aligned}$$

(1.4.94)

and

$$\begin{aligned} \Vert e^{ \tau \varDelta }\nabla \cdot \varphi \Vert _{L^\infty (\varOmega )}\le C_2\tau ^{-\frac{1}{2}-\frac{n}{2p}}\Vert \varphi \Vert _{L^p(\varOmega )}~~\hbox {for all}~~ \varphi \in C^1(\varOmega ;\mathbb {R}^n). \end{aligned}$$

(1.4.95)

By (1.4.79) and (1.4.80), there exists $t_0>3$ such that

$$\begin{aligned} 24 C_2( \chi \Vert \nabla w(\cdot ,t)\Vert _{L^3(\varOmega )}+ \Vert u (\cdot ,t)\Vert _{L^3(\varOmega )} )\le 1~~\hbox { for all} ~~t>t_0-1. \end{aligned}$$

(1.4.96)

Now in order to prove the lemma, it is sufficient to derive a bound, independent of $T\in (t_0,\infty )$, for $M(T)\triangleq \displaystyle \sup _{t_0-1<t<T}\{t\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}\}$.

By the variation-of-constants representation of n, we have

$$\begin{aligned} \begin{aligned} n(\cdot ,t)&= e^{\varDelta }n(\cdot ,t-1)+\displaystyle \chi \!\int _{t-1}^t e^{(t-s)\varDelta }\nabla \cdot (n\nabla w)(\cdot ,s)ds- \int _{t-1}^t e^{(t-s)\varDelta }(u\cdot \nabla n)(\cdot ,s)ds\\&- \displaystyle \mu \int _{t-1}^t e^{(t-s)\varDelta } n^2(\cdot ,s) ds. \end{aligned} \end{aligned}$$

(1.4.97)

Since $e^{ (t-s)\varDelta }$ is nonnegative in $\varOmega$ for all $0<s<t$ due to the maximum principle, it follows from the nonnegativity of n that for all $t\in (t_0, T)$

$$\begin{aligned}&\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}\\ \le&\Vert e^{\varDelta }n(\cdot ,t-1)\Vert _{L^\infty (\varOmega )}+\chi \!\int _{t-1}^t \Vert e^{(t-s)\varDelta }\nabla \cdot (n\nabla w)(\cdot ,s)\Vert _{L^\infty (\varOmega )} ds\\&+ \int _{t-1}^t \Vert e^{(t-s)\varDelta }(u\cdot \nabla n)(\cdot ,s)\Vert _{L^\infty (\varOmega )} ds \end{aligned}$$

which along with (1.4.94)–(1.4.96) and (1.4.2) yields

$$\begin{aligned}&\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}\\ \le&C_1\Vert n(\cdot ,t-1)\Vert _{L^1(\varOmega )}+C_2\chi \!\int _{t-1}^t (t-s)^{-\frac{5}{6}}\Vert (n\nabla w)(\cdot ,s)\Vert _{L^3(\varOmega )} ds\\&+C_2\!\int _{t-1}^t (t-s)^{-\frac{5}{6}}\Vert (u n)(\cdot ,s)\Vert _{L^3(\varOmega )} ds\\ \le&\displaystyle \frac{C_1 |\varOmega |}{\mu (t-1+\gamma )}+ \frac{6C_2}{t-1} (\chi \displaystyle \max _{t_0-1<s<T}\Vert \nabla w(\cdot ,s)\Vert _{L^3 (\varOmega )} + \Vert u (\cdot ,t)\Vert _{L^2(\varOmega )} ) \! \cdot M(T)\\ \le&\displaystyle \frac{C_1 |\varOmega |}{\mu (t-1+\gamma )}+ \frac{1}{ 4 (t-1)} M(T). \end{aligned}$$

Hence,

$$ M(T)\le \displaystyle \frac{4C_1 |\varOmega |}{\mu }+2\displaystyle \sup _{t_0-1<s<t_0}\{s\Vert n(\cdot ,s)\Vert _{L^\infty (\varOmega )}\}, $$

which readily yields (1.4.93) since $T>t_0$ is arbitrary, and thus ends the proof.

In light of Lemma 1.46, we can derive a pointwise estimate c(x, t) from below.

Lemma 1.47

Let (n, w, u) be the global classical solution of (1.4.1) obtained in Thenrem 1.2 with $r=0, \mu >0$. Then there exists $\kappa >0$ fulfilling

$$\begin{aligned} c(x,t)\ge \frac{\displaystyle \inf _{x\in \varOmega }c_0(x)}{(t+1)^{\kappa }}. \end{aligned}$$

(1.4.98)

Proof

By the second equation of (1.4.1) and Lemma 1.46, we can see that

$$w_t\le \triangle w- |\nabla w|^2 +\frac{C_1}{t+1}-u \cdot \nabla w $$

with some $C_1>0$ for all $t>0$. Let $y\in C^1([0,\infty ))$ denote the solution of the initial-value problem $y'(t)=\frac{C_1}{t+1}$, $y(0)=\Vert w_0\Vert _{L^\infty (\varOmega )}$, then from the comparison principle, we infer that

$$\begin{aligned} w(x,t)\le \Vert w_0\Vert _{L^\infty (\varOmega )}+C_1\ln (t+1) ~~\hbox { for all} ~t>0, \end{aligned}$$

(1.4.99)

which along with $w= -\ln (\frac{c}{\Vert c_0\Vert _{L^\infty (\varOmega )}})$, yields (1.4.98) with $\kappa =C_1$.

Now utilizing the decay information on $|\nabla w|$ in $L^\infty (\varOmega )$ in (1.4.77) again, and thanks to the precise information on the decay of $\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ in Lemma 1.46, we can obtain the desired estimate for $\Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ from below as well as the upper estimate for $\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}$.

Lemma 1.48

Let (n, w, u) be the solution of (1.4.1) obtained in Thenrem 1.2 with $r=0, \mu >0$. Then one can find $C_1>0$ and $C_2>0$ fulfilling

$$\begin{aligned} \Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}\ge \frac{1}{|\varOmega |}\Vert n(\cdot ,t)\Vert _{L^1(\varOmega )} \ge \frac{C_1}{t+1} ~~~\hbox { for all}~~t>0 \end{aligned}$$

(1.4.100)

as well as

$$\begin{aligned} \Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \frac{C_2}{t+1}~~\hbox {for all}~~t>0. \end{aligned}$$

(1.4.101)

Proof

We first adapt the method in Lemma 1.46 to derive the precise decay rate of $\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}$. By (1.4.79) and (1.4.80), one can choose some $t_0>2$ such that

$$\begin{aligned} 4 c_1\int ^{\infty }_0 (1+\sigma ^{-\frac{1}{2}}) e^{-\lambda _1 \sigma } d\sigma (\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}+ \Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )})\le 1 \end{aligned}$$

(1.4.102)

for all $t>\frac{t_0}{2}$ and then let $M(T)\triangleq \displaystyle \sup _{\frac{t_0}{2}<s<T}\{s\Vert \nabla w(\cdot ,s)\Vert _{L^\infty (\varOmega )}\}$ for all $T>t_0$.

By the variation-of-constants representation of w, we have

$$ w(\cdot ,t)= e^{\frac{t}{2} \varDelta }w(\cdot ,\frac{t}{2})-\!\int _{\frac{t}{2}}^t e^{(t-s)\varDelta }|\nabla w|^2(\cdot ,s)ds+ \int _{\frac{t}{2}}^t e^{(t-s)\varDelta }( n-u\cdot \nabla w) (\cdot ,s)ds $$

for all $t_0<t<T$. We then show that

$$\begin{aligned}&\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )} \\ \le&\Vert \nabla e^{\frac{t}{2} \varDelta }w(\cdot ,\frac{t}{2})\Vert _{L^\infty (\varOmega )} +\!\int _{\frac{t}{2}}^t\Vert \nabla e^{(t-s)\varDelta }|\nabla w|^2\Vert _{L^\infty (\varOmega )}+ \int _{\frac{t}{2}}^t \Vert \nabla e^{(t-s)\varDelta } n\Vert _{L^\infty (\varOmega )} \\&+ \int _{\frac{t}{2}}^t \Vert \nabla e^{(t-s)\varDelta } (u\cdot \nabla w)\Vert _{L^\infty (\varOmega )} \\ \le&c_1(1+t^{-\frac{1}{2}})e^{-\frac{\lambda _1t}{2}} \Vert w(\cdot ,\frac{t}{2})\Vert _{L^\infty (\varOmega )}+ c_1\int _{\frac{t}{2}}^t(1+(t-s)^{-\frac{1}{2}})e^{-\lambda _1(t-s)} \Vert n(\cdot ,s)\Vert _{L^\infty (\varOmega )} \\&+c_1\int _{\frac{t}{2}}^t(1+(t-s)^{-\frac{1}{2}})e^{-\lambda _1(t-s)} \Vert \nabla w(\cdot ,s)\Vert _{L^\infty (\varOmega )} (\Vert \nabla w(\cdot ,s)\Vert _{L^\infty (\varOmega )}+\Vert u(\cdot ,s)\Vert _{L^\infty (\varOmega )}) \\ \le&c_1(1+t^{-\frac{1}{2}})e^{-\frac{\lambda _1t}{2}}(\Vert w_0\Vert _{L^\infty (\varOmega )}+c_2\ln (t+1))+ \frac{2c_1 c_2}{t} \int ^{\infty }_0 (1+\sigma ^{-\frac{1}{2}}) e^{-\lambda _1 \sigma }d\sigma \\&+ \frac{2c_1}{t}\int ^{\infty }_0 (1+\sigma ^{-\frac{1}{2}}) e^{-\lambda _1 \sigma }d\sigma \displaystyle \sup _{t\ge \frac{t_0}{2}} (\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}+\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )})\cdot M(T) \\ \le&c_1(1+t^{-\frac{1}{2}})e^{-\frac{\lambda _1t}{2}}(\Vert w_0\Vert _{L^\infty (\varOmega )}+c_2 \ln (t+1))+ \frac{2c_1 c_2}{t} \int ^{\infty }_0 (1+\sigma ^{-\frac{1}{2}}) e^{-\lambda _1 \sigma }d\sigma \\&+ \frac{1}{2t} M(T) \end{aligned}$$

by using Lemma 1.4.1(i), (1.4.99), (1.4.93) and (1.4.102). This along with the definition of M(T) yields

$$ M(T)\le 2\displaystyle \sup _{\frac{t_0}{2}<s<t_0}\{s\Vert \nabla w(\cdot ,s)\Vert _{L^\infty (\varOmega )}\} + c_4 $$

with some constant $c_4>0$ as $\displaystyle \lim _{t\rightarrow \infty } t\ln (t+1) e^{-\lambda _1 t}=0$. Hence, upon the definition of M(T), we arrive at (1.4.101) with an evident choice of $C_2$.

Continuing with the proof, we claim that there exists $c_4>0$ such that

$$\begin{aligned} \Vert n(\cdot ,t)\Vert _{L^\infty (\varOmega )}\ge \frac{1}{|\varOmega |}\Vert n(\cdot ,t)\Vert _{L^1(\varOmega )} \ge \frac{c_4}{t+1} ~~~\hbox { for all}~~t>0. \end{aligned}$$

(1.4.103)

Indeed, from the $n-$equation of (1.4.1) with $r=0$ and Young’s inequality, it follows that $\displaystyle \frac{d}{dt}\int _\varOmega \ln n = \displaystyle \int _\varOmega \frac{ |\nabla n|^2}{n^2} +\chi \displaystyle \int _\varOmega \frac{1}{n} \nabla \cdot (n\nabla w) - \mu \displaystyle \displaystyle \int _\varOmega n \ge -\frac{\chi ^2}{4} \displaystyle \int _\varOmega |\nabla w|^2 - \mu \displaystyle \displaystyle \int _\varOmega n.$ Inserting (1.4.2) and (1.4.101) into the above inequality yields $\frac{d}{dt}\int _\varOmega \ln n \ge -\frac{\chi ^2}{4} \frac{C_2^2|\varOmega |}{(t+1)^2} -\frac{|\varOmega |}{t+\gamma }$ and thus

$$\begin{aligned} \displaystyle \int _\varOmega \ln n(\cdot ,t) \ge -|\varOmega |\ln (t+\gamma )-c_5 ~~~\hbox { for all}~~t>1 \end{aligned}$$

(1.4.104)

with some $c_5>0$. On the other hand, by the Jensen inequality, we have

$$\begin{aligned} |\varOmega |\ln (\displaystyle \int _\varOmega n(\cdot ,t))-|\varOmega |\ln |\varOmega |=|\varOmega | \ln \{ \frac{1}{|\varOmega |}\int _\varOmega n(\cdot ,t))\} \ge \displaystyle \int _\varOmega \ln n(\cdot ,t). \end{aligned}$$

This inequality together with (1.4.104) readily leads to (1.4.100).

With the above lemmas at hand, we can now complete the proof of Theorem 1.4.

Proof of Theorem1.4. By $w= -\ln (\frac{c}{\Vert c_0\Vert _{L^\infty (\varOmega )}})$, Lemma 1.46 and Lemma 1.48, one can see that $(n,\frac{|\nabla c|}{c})\longrightarrow (0,0)$ in $L^\infty (\varOmega )$ algebraically as $t\rightarrow \infty$. Hence, it suffices to show the decay property of c(x, t). In view of the w-equation in (1.4.1), (1.4.103), (1.4.101) and $\nabla \cdot u=0$, we can pick $C_i>0$ $(i=1,2,3)$ such that

$$ \displaystyle \frac{d}{dt}\int _\varOmega w =\displaystyle \int _\varOmega n -\displaystyle \int _\varOmega |\nabla w|^2- \int _\varOmega u \cdot \nabla w \ge \frac{C_1|\varOmega |}{t+1} - \frac{C_2|\varOmega |}{(t+1)^2},~ $$

and hence $\int _\varOmega w(\cdot ,t)\ge C_1|\varOmega |\ln (t+1)-C_3,$ which entails that for any $t>0$ there exists $x_0(t)\in \varOmega$ such that $w(x_0(t),t)\ge C_1\ln (t+1)-\frac{C_3}{|\varOmega |}.$ Since for each $\varphi \in W^{1,p}(\varOmega )$ with $p>2$, there exists $C_4>0$ such that

$$|\varphi (x)-\varphi (y)|\le C_4 |x-y|^{1-\frac{2}{p}} \Vert \nabla \varphi \Vert _{L^p(\varOmega )} ~~\hbox { for all}~x,y\in \varOmega , $$

we therefore obtain from (1.4.101) that

$$\begin{aligned} w(x,t)&\ge w(x_0(t),t)-|x-x_0(t)|\Vert \nabla w(\cdot ,t)\Vert _{L^\infty (\varOmega )}\\&\ge C_1\ln (t+1)-\frac{C_3}{|\varOmega |}-C_4 \hbox {diam}(\varOmega ),\nonumber \end{aligned}$$

(1.4.105)

and thereby $c(x,t)\le \displaystyle \frac{C_5}{(t+1)^{C_1}}$ for $x\in \varOmega , t>0$ for some $C_5>0$. This together with (1.4.98) shows that c(x, t) actually converges to 0 in $L^\infty (\varOmega )$ algebraically as $t\rightarrow \infty$, and thus ends the proof of Theorem 1.4.