Density-Suppressed Motility System

Abstract

The reaction–diffusion models can reproduce a wide variety of exquisite spatio-temporal patterns arising in embryogenesis, development and population dynamics due to the diffusion-driven (Turing) instability [102, 162].

5.1 Introduction

The reaction–diffusion models can reproduce a wide variety of exquisite spatio-temporal patterns arising in embryogenesis, development and population dynamics due to the diffusion-driven (Turing) instability (Kondo and Miura 2010; Murray 2001). Many of them invoke nonlinear diffusion enhanced by the local environment condition to accounting for population pressure (cf. Méndez et al. 2012), volume exclusion (cf. Painter and Hillen 2002; Wang and Hillen 2007) or avoidance of danger (cf. Murray 2001) and so on. However, the opposite situation where the species will slow down its random diffusion rate when encountering external signals such as the predator in pursuit of the prey (Jin and Wang 2021; Kareiva and Odell 1987) and the bacterial searching food (Keller and Segel 1970, 1971b) has not been considered. Recently, a so-called “self-trapping” mechanism was introduced in Liu (2011) by a synthetic biology approach onto programmed bacterial Escherichia coli cells which excrete signaling molecules acyl-homoserine lactone (AHL) such that at low AHL levels, the bacteria undergo run-and-tumble random motion and are motile, while at high AHL levels, the bacteria tumble incessantly and become immotile due to the vanishing macroscopic motility. Remarkably, Escherichia coli cells formed the outward expanding ring (strip) patterns in the petri dish (Fig. 5.1).

To understand the underlying patterning mechanism, both two-component and three-component “density-suppressed motility” reaction–diffusion systems are proposed. In this chapter, we study the global existence, boundedness, asymptotic behavior of solutions and the existence of traveling wave solutions for density-suppressed motility models. The chapter is divided into two parts. Section 5.3 is devoted to investigate a two-component density-suppressed motility model and shows the existence of traveling wave solutions which are genuine patterns observed in the experiment of Liu (2011), whereas Sect. 5.4 shows the existence and the asymptotic behavior of global weak solution for a three-component quasilinear density-suppressed motility model.

Fig. 5.1

The figure is taken from Fig. 1 in Liu (2011) for illustration

Time-lapsed photographs of spatio-temporal patterns formed by the engineered Escherichia coli strain CL3 (see details in Liu 2011).

Chemotaxis plays an outstanding role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites (Isenbach 2004; Murray 2001). The celebrated mathematical model describing chemotactic migration processes at population level is the Keller–Segel system of the form

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\nabla \cdot (\gamma (u,v)\nabla u-u\phi (u,v)\nabla v),&x\in \varOmega ,t>0,\\&v_{t}=d\varDelta v-v+u,&x\in \varOmega ,t>0, \end{aligned} \right. \end{aligned}$$

(5.1.1)

in a bounded domain $\varOmega \subset \mathbb {R}^n$ where $u = u(x, t)$ denotes the population density and $v=v(x,t)$ is the concentration of chemical substance secreted by the population itself (Keller and Segel 1970). The prominent feature of (5.1.1) is the ability of the constitutive ingredient cross-diffusion thereof to describe the collective behavior of cell populations mediated by a chemoattractant. Indeed, a rich literature has revealed that the Neumann initial-boundary value problem for the classical Keller–Segel system

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\triangle u- \nabla \cdot (u\nabla v),\quad&x\in \varOmega ,t>0,\\&v_{t}=d\varDelta v-v+u,&x\in \varOmega ,t>0 \end{aligned} \right. \end{aligned}$$

(5.1.2)

possesses solutions blowing up in finite time with respect to the spatial $L^\infty$ norm of u in two- and even higher dimensional frameworks under some condition on the mass and the moment of the initial data (Herrero and Velázquez 1996, 1997; Winkler 2013, see also the surveys Bellomo et al. 2016). Apart from that, when $\phi$ and $\gamma$ in (5.1.1) are only smooth positive functions of u on $[0, \infty )$, a considerable literature underlines the crucial role of asymptotic beahvior of the ratio $\frac{\gamma (u)}{\phi (u)}$ at large values of u with regard to the occurrence of singularity phenomena (see recent progress in Ishida et al. 2014; Winkler 2017c, 2019e).

As a simplification of (5.1.1), the Keller–Segel system with density-dependent motility

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\nabla \cdot (\gamma (v)\nabla u-u\phi (v)\nabla v),&x\in \varOmega ,t>0,\\&v_{t}=d\varDelta v-v+u,&x\in \varOmega ,t>0 \end{aligned} \right. \end{aligned}$$

(5.1.3)

was proposed to describe the aggregation phase of Dictyostelium discoideum (Dd) cells in response to the chemical signal cyclic adenosine monophosphate (cAMP) secreted by Dd cells in Keller and Segel (1971b). Here, the signal-dependent diffusivity $\gamma (v)$ and chemotactic sensitivity function $\phi (v)$ are linked through

$$ \phi (v) = (\alpha -1)\gamma '(v), $$

where $\alpha \ge 0$ denotes the ratio of effective body length (i.e., distance between the signal–receptors) to the walk length (see Cai et al. 2022 for details). Notice that when $\alpha =0$, there is only one receptor in a cell, and hence, chemotaxis is driven by the indirect effect of chemicals in the absence of the chemical gradient sensing. In this case, (5.1.3) reads as

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\triangle (\gamma (v)u),\quad&x\in \varOmega ,t>0,\\&v_{t}=d\varDelta v-v+u,&x\in \varOmega ,t>0, \end{aligned} \right. \end{aligned}$$

(5.1.4)

where the considered diffusion process of the population is essentially Brownian, and the assumption $\gamma '(v)<0$ accounts for the repressive effect of the chemical concentration on the population motility (Fu et al. 2012). In the context of acyl-homoserine lactone (AHL) density-dependent motility, the extended model of (5.1.4)

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\varDelta (u\gamma (v)) +\beta \displaystyle \frac{uw^2}{w^2+\lambda },&x\in \varOmega ,~t>0, \\&v_{t}=D\varDelta v+u-v,&x\in \varOmega ,~t>0,\\&w_{t}=\varDelta w-\displaystyle \frac{uw^2}{w^2+\lambda },&x\in \varOmega ,~t>0 \end{aligned} \right. \end{aligned}$$

(5.1.5)

was proposed in Liu (2011) to advocate that spatio-temporal pattern of Escherichia coli cells can be induced via so-called “self-trapping” mechanisms, that is, at low AHL levels, the bacteria undergo run-and-tumble random motion, while at high AHL levels, the bacteria tumble incessantly and become immotile at the macroscale.

In comparison with plenty of results on the Keller–Segel system where the diffusion depends on the density of cells, the respective knowledge seems to be much less complete when the cell dispersal explicitly depends on the chemical concentration via the motility function $\gamma (v)$, which is due to considerable challenges of the analysis caused by the degeneracy of $\gamma (v)$ as $v\rightarrow \infty$ from the mathematical point of view. Indeed, to the best of our knowledge, Yoon and Kim (2017) showed that in the case of $\gamma (v)=\frac{c_0}{v^k}$ for small $c_0$, problem (5.1.4) admits a global classical solutions in any dimensions. The smallness condition on $c_0$ is removed lately in Ahn and Yoon (2019) for the parabolic–elliptic version of (5.1.4) with $0<k < \frac{n}{(n-2)_+}$. Furthermore, for the full parabolic system (5.1.4) in the three-dimensional setting, Tao and Winkler (2017a) showed the existence of certain global weak solutions, which become eventually smooth and bounded for suitably small initial data $u_0$ under the assumption

$$\begin{aligned}&(H)~~\gamma (v)\in C^3([0,\infty )),~\text {and there exist } \gamma _1,~\gamma _2,\eta >0 \text { such that }0<\gamma _1\le \gamma (v)\le \gamma _2,\nonumber \\&|\gamma ^{\prime }(v)|<\eta \text { for all }v\ge 0. \end{aligned}$$

It should be remarked that based on the comparison method, Fujie and Jiang (2021) obtained the uniform-in-time boundedness to (5.1.4) in two-dimensional setting for the more general motility function $\gamma$ and in the three-dimensional case under a stronger growth condition on $1/\gamma$, respectively. In addition, they investigated the asymptotic behavior to the parabolic–elliptic analog of (5.1.4) under the assumption $\displaystyle \max _{0\le v< +\infty }\frac{|\gamma '(v)|^2}{\gamma (v)}<+\infty$ or $\gamma (v)=v^{-k}$ with $0<k<\frac{n}{(n-2)_+}$ in Fujie and Jiang (2020) and Jiang and Laurençot (2021).

On the considered time scales of cell migration, e.g., metastatic cells moving in semi-solid medium, often it is relevant to take into account the growth of the population. A prototypical choice to accomplish this is the addition of logistic growth terms $\kappa u-\mu u^2$ in the cell equation (Murray 2001). From the mathematical point of view, the dissipative action of logistic-like growth possibly prevents the occurrence of singularity phenomena in various chemotaxis models. For instance, for the chemotaxis-growth system (Fu et al. 2012)

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\triangle (\gamma (v)u)+u(a-bu),\quad&x\in \varOmega ,t>0,\\&v_{t}=\varDelta v-v+u,&x\in \varOmega ,t>0, \end{aligned} \right. \end{aligned}$$

(5.1.6)

it is shown in Jin et al. (2018) that in two-dimensional setting, the system admits a unique global classical solution if the motility function $\gamma \in C^3([0,\infty ))$ satisfies $\gamma (v) > 0$ and $\gamma '(v) < 0$ for all $v\ge 0$, $\lim _{v\rightarrow \infty } \gamma (v)=0$ and $\displaystyle \lim _{v\rightarrow \infty }\frac{\gamma ^{\prime }(v)}{\gamma (v)}$, and even the constant steady state (1, 1) is globally asymptotically stable if $a=b>\frac{1}{16}\displaystyle \max _{0\le v< +\infty } \frac{|\gamma '(v)|^2}{\gamma (v)}$. For $a=b$, the global existence thereof in the higher dimensions has been proved for large a and b (Wang and Wang 2019a), while for small a and b, the respective model can generate pattern formation (see Ma et al. 2020). The reader is referred to Lv and Wang (2020, 2021) for the other studies on the related variants involving super-quadratic degradation terms.

As recalled above, the existing results for (5.1.6) are confined to the global well-posedness and asymptotic behaviors of solutions and stationary solutions (pattern formation). However, the traveling wave solutions, which are genuinely relevant to the experiment observation of Liu (2011), are not investigated mathematically except for a special case that $\gamma (v)$ is piecewise constant. When $\gamma (v)$ is a constant, equations of (5.1.6) are decoupled each other and the first equation becomes the well-known Fisher-KPP equation—a benchmark model for the study of traveling wave solutions of reaction–diffusion equations (Murray 2001). However, once $\gamma (v)$ is non-constant, (5.1.6) becomes a coupled system with cross-diffusion, and the study of traveling wave solutions drastically becomes difficult.

The purpose of Sect. 5.3 is to make some progress in this direction and explore the existence of traveling wave solutions to (5.1.6) with allowable wave speeds. With general $\gamma (v)$, the analysis and results will be too complicated to have an elegant presentation. Noticing that the key feature of $\gamma (v)$ lies in the monotone property $\gamma '(v)<0$, in this Section, we consider a general algebraically decreasing motility function

$$\begin{aligned} \gamma (v)=\frac{1}{(1+v)^m}, \ m>0. \end{aligned}$$

(5.1.7)

However, our argument can be directly extended to other forms of motility function, such as the exponentially decreasing $\gamma (v)=e^{-\chi v}$ and so on.

To put things in perspective, we rewrite (5.1.6) as

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (\gamma (v)\nabla u+u\gamma '(v)\nabla v)+u(a-bu), \\&v_t=\varDelta v+u-v, \end{aligned} \right. \end{aligned}$$

(5.1.8)

which is a Keller–Segel-type chemotaxis model proposed in Keller and Segel (1971b) with growth. For the classical chemotaxis-growth system

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (\nabla u-\chi u\nabla v)+u(a-bu), \\&\tau v_t=\varDelta v+u-v, \end{aligned} \right. \end{aligned}$$

(5.1.9)

traveling wave solutions are investigated in a series of works (Nadin et al. 2008; Salako and Shen 2017a, b, 2018, 2020) for both cases $\tau =0$ and $\tau =1$, where $\chi >0$ denotes the chemotactic coefficient. The existence of traveling wave solutions with minimal wave speed depending on a and $\chi$ was obtained, and the asymptotic wave speed as $\chi \rightarrow 0$ as well as the spreading speed were examined in detail in Salako and Shen (2017a, 2017b, 2018) and Salako et al. (2019) where the major tool used therein to prove the existence of traveling wave solutions is the parabolic comparison principle. Except traveling wave solutions, the chemotaxis-growth system (5.1.9) can also drive other complex patterning dynamics (cf. Kolokolnikov et al. 2014; Ma et al. 2012; Painter and Hillen 2011). When the volume filling effect is considered in (5.1.9) (i.e., $\chi u\nabla v$ is changed to $\chi u(1-u)\nabla v$), the traveling wave solutions with minimal wave speed were shown to exist in Ou and Yuan (2009) for small chemotactic coefficient $\chi >0$. For the original singular Keller–Segel system generating traveling waves without cell growth, we refer to Keller and Segel (1971a), Li et al. (2014), Wang (2013) and references therein. In contrast to the classical chemotaxis-growth system (5.1.9), both diffusive and chemotactic coefficients in the system (5.1.8) are non-constant. This not only makes the analysis more complex, but also makes the parabolic comparison principle inapplicable due to the nonlinear diffusion. In this section, we shall develop some new ideas to tackle the various difficulties induced by the nonlinear motility function $\gamma (v)$ and establish the existence of traveling wave solutions to (5.1.6).

In Sect. 5.3, we shall establish the existence of traveling wave solutions and wave speed of (5.1.6) and explore how the density-suppressed motility influences traveling wave profiles and “the minimal wave speed”. In the spatially homogeneous situation, the steady states are (0, 0) and (a/b, a/b), which are, respectively, unstable (saddle point) and stable node. This suggests that we should look for traveling wavefront solutions to (5.1.6) connecting (a/b, a/b) to (0, 0). Moreover, negative u and v have no physical meanings to what we have in mind in the sequel.

A nonnegative solution (u(x, t), v(x, t)) is called a traveling wave solution of (5.1.6) connecting (a/b, a/b) to (0, 0) and propagating in the direction $\xi \in S^{N-1}$ with speed c if it is of the form

$$(u(x,t),v(x,t))=(U(x\cdot \xi -ct),V(x\cdot \xi -ct))=: (U(z), V(z))$$

satisfying the following equations:

$$\begin{aligned} \left\{ \begin{aligned}&(\gamma (V)U)''+cU'+U(a-bU)=0,\\&V''+cV'+U-V=0 \end{aligned} \right. \end{aligned}$$

(5.1.10)

and

$$\begin{aligned} (U(-\infty ),V(-\infty ))=(a/b,a/b), \ (U(+\infty ),V(+\infty ))=(0,0), \end{aligned}$$

(5.1.11)

where $'=\frac{d}{dz}$. In the first part of this chapter, we proceed to find the constraints on the parameters to exclude the spatio-temporal pattern formation and guarantee the existence of traveling wave solutions connecting the two constant steady states.

Denoting

$$\begin{aligned} b^*(m,a)=\max \left\{ 9m,3m+2\sqrt{\frac{m(m+1)a}{1+a}}\right\} , \end{aligned}$$

(5.1.12)

we obtain the following two theorems (Li and Wang 2021c).

Theorem 5.1

Let $\gamma (v)$ be given in (5.1.7). Then for any $c\ge 2\sqrt{a}$ and $b> b^*(m,a)$, the system (5.1.6) has a traveling wave solution $(u(x,t),v(x,t))=(U(x\cdot \xi -ct),V(x\cdot \xi -ct))$ with speed c in the direction $\xi \in S^{N-1}$ for all $(x,t)\in \mathbb R^N\times [0,+\infty )$, satisfying

$$\begin{aligned} \lim _{z\rightarrow +\infty }\frac{U(z)}{e^{-\lambda z}}=1,\quad \lim _{z\rightarrow +\infty }\frac{V(z)}{e^{-\lambda z}}=\frac{1}{1+a} \end{aligned}$$

(5.1.13)

with $\lambda =\frac{c-\sqrt{c^2-4a}}{2}$ and

$$\begin{aligned} \liminf _{z\rightarrow -\infty }U(z)>0\quad \hbox {and}\quad \liminf _{z\rightarrow -\infty }V(z)>0. \end{aligned}$$

Moreover, if

$$\begin{aligned} \mathscr {K}(m,a)=m\sqrt{\frac{a(1+a)}{m(m+1)}}\left( \sqrt{\frac{a(1+a)}{m(m+1)}}+1\right) ^m<1, \end{aligned}$$

(5.1.14)

we have

$$ \lim _{z\rightarrow -\infty }U(z)=\lim _{z\rightarrow -\infty }V(z)=a/b $$

and

$$\lim _{z\rightarrow \pm \infty }U'(z)=\lim _{z\rightarrow \pm \infty }V'(z)=0.$$

Theorem 5.2

For $c<2\sqrt{a}$, there is no traveling wave solution $(u(x,t),v(x,t))=(U(x\cdot \xi -ct),V(x\cdot \xi -ct))$ of (5.1.6) connecting the constant solutions $(a/b,a/b)$ and (0, 0) with speed c.

Remark 5.1

Theorems 5.1 and 5.2 imply that $c=2\sqrt{a}$ is the minimal wave speed same as the one for the classical Fisher-KPP equation and irrelevant to the decay rate of the motility function. Different from the Fisher-KPP equation, a lower bound $b^*(m,a)$ for b is induced by the density-suppressed motility. As $m\rightarrow 0$, $\gamma (v) \rightarrow 1$ and the equation for u becomes the classical Fisher-KPP equation. Noticing

$$\lim _{m\rightarrow 0} b^*(m,a)=0 \ \text {and} \ \lim _{m\rightarrow 0} \mathscr {K}(m,a)\rightarrow 0,$$

our result well agrees with that for the classical Fisher-KPP equation.

Proof strategies for Theorems 5.1 and5.2.  Since the model (5.1.6) is a cross-diffusion system, see also (5.1.8), many classical tools proving the existence of traveling waves such as phase plane analysis, topological methods and bifurcation analysis (cf. Volpert et al. 1994), among others, become infeasible. Motivated from excellent works of Salako and Shen (2017a, 2018, 2020) for the chemotaxis-growth model (5.1.9) by constructing super- and sub-solutions and proving the existence of traveling wave solutions as the large time limit of solutions in the moving-coordinate system based on the parabolic comparison principle, we plan to achieve our goals in a similar spirit. However, substantial differences exist between the models (5.1.6) and (5.1.9). The nonlinear motility function $\gamma (v)$ in (5.1.6) refrains us from employing the parabolic comparison principle and constructing super- and sub-solutions with the same decay rate at the far field, which are crucial ingredients used for (5.1.9) in Salako and Shen (2017a, 2018). In the first part of this chapter, we develop two innovative ideas to overcome these barriers. First, we introduce an auxiliary parabolic problem (5.3.16) with constant diffusion to which the method of super- and sub-solutions applies (see Sect. 5.3.2). This auxiliary problem subtly bypasses the barriers induced by the nonlinear diffusion but its time-asymptotic limit yields a solution to an elliptic problem (5.3.29) whose fixed points indeed correspond to solutions to (5.1.10)—namely traveling wave solutions to our concerned system (5.1.6) (see Sect. 5.3.3). Second, we construct a sequence of relaxed sub-solution $\underline{U}_n(x)$ for any $n>1$ with a spatially inhomogeneous decay rate $\theta _1(x)$ which approaches to the constant decay rate of the super-solution $\overline{U}(x)$ as $x \rightarrow +\infty$ (see Sect. 5.2). With them, we use the method of super- and sub-solutions to construct solutions to the auxiliary parabolic problem (5.3.16) in appropriate function space and manage to show its time-asymptotic limit problem has a fixed point. This is a fresh idea substantially different from the works (Salako and Shen 2017a, 2018) where the super- and sub-solutions were directly constructed with the same decay rates by taking the advantage of constant diffusion.

We divide the proof of Theorem 5.1 into four steps. In step 1, we construct an auxiliary parabolic problem (5.3.16) with constant diffusion and prove its global boundedness uniformly in time (see Proposition 5.1) by the method of super- and sub-solutions. In step 2, we show that the limit of global solutions to (5.3.16) as $t \rightarrow \infty$ yields a semi-wavefront solution to an elliptic problem (5.3.29) with some compactness argument (see Proposition  5.2). In step 3, we show that the solution obtained in step 2 satisfies the boundary condition (5.1.11) by direct estimates under some constraints on m and a (see Proposition 5.3), which hence warrants that the semi-wavefront solution is indeed a wavefront solution in $\mathbb {R}$. Finally, in step 4, we use Schauder’s fixed point theorem to prove that (5.3.29) has a fixed point which gives a solution to (5.1.10) in $\mathbb {R}$ satisfying (5.1.11) (see Sect. 5.3.3), where the trick of utilizing relaxed sub-solution $\underline{U}_n(x)$ with spatially inhomogeneous decay rate is critically used to obtain the continuity of the solution map. Theorem 5.2 is proved directly by an argument of contradiction.

Section 5.4 is devoted to the asymptotic behavior of a quasilinear Keller–Segel system with signal-suppressed motility. In the context of the diffusion of cells in a porous medium (see the discussions in Calvez and Carrillo 2006; Vázquez 2007), Winkler (2020) considered the cross-diffusion system

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\triangle (\gamma (v)u^m),\quad&x\in \varOmega ,t>0,\\&v_{t}=\varDelta v-v+u,&x\in \varOmega ,t>0 \end{aligned} \right. \end{aligned}$$

(5.1.15)

in smoothly bounded convex domains $\varOmega \subset \mathbb {R}^n$, where $m> 1$, $\gamma$ generalizes the prototype $\gamma (v) =a+b(v+d)^{-\alpha }$ with $a\ge 0,b>0,d\ge 0$ and $\alpha \ge 0$, and proved the boundedness of global weak solutions to the associated initial-boundary value problem under some constriction on m and $\alpha$, which particularly indicates that increasing m in the cell equation goes along with a certain regularizing effect despite both the diffusion and the cross-diffusion mechanisms implicitly contained in (5.1.15) are simultaneously enhanced.

In a recent paper (Jin et al. 2020), Jin et al. considered the three-component system

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\varDelta (\gamma (v)u) +\beta uf(w)-\theta u,&x\in \varOmega ,~t>0,\\&v_{t}=D\varDelta v+u-v,&x\in \varOmega ,~t>0,\\&w_{t}=\varDelta w-uf(w),&x\in \varOmega ,~t>0 \end{aligned} \right. \end{aligned}$$

(5.1.16)

in a bounded domain $\varOmega \subset \mathbb {R}^2$, where $\beta$, $D>0$ and $\theta \ge 0$, the random motility function $\gamma (v)$ satisfies (H) and functional response function f(w) fulfills the assumption

$$\begin{aligned} f(w)\in C^1([0,\infty )),~f(0)=0, ~ f(w)>0~\text {in}~(0,\infty )~ \text {and}~f'(w)>0~\text {on}~[0,\infty ). \end{aligned}$$

(5.1.17)

Based on the method of energy estimates and the Moser iteration, they showed the uniform boundedness to initial-boundary value problem of (5.1.16), inter alia the asymptotic behavior thereof when parameter D is suitably large. Note that the authors of Lv and Wang (2022) showed the existence of global classical solutions to system (5.1.16) without the restriction (H) on $\gamma (v)$. In synopsis of the above results, one natural problem seems to consist in determining to which extent nonlinear diffusion of porous medium type may influence the solution behavior in chemotaxis systems involving density-suppressed motility. Accordingly, the purpose of the present work is to address this question in the context of the particular choice $\gamma (v)=v^{-\alpha }$ with $\alpha >0$ instead of assumption (H) in (5.1.16). Specifically, we consider the asymptotic behavior to the initial-boundary value problem

$$\begin{aligned} \left\{ \begin{aligned}&u_{t}=\varDelta (\displaystyle \frac{u^m}{v^\alpha }) +\beta uf(w),&x\in \varOmega ,~t>0, \\&v_{t}=D\varDelta v+u-v,&x\in \varOmega ,~t>0,\\&w_{t}=\varDelta w-uf(w),&x\in \varOmega ,~t>0 \end{aligned} \right. \end{aligned}$$

(5.1.18)

along with the initial conditions

$$\begin{aligned} u(x,0)=u_0, v(x,0)=v_0~\hbox {and}~ w(x,0)=w_0, ~~x\in \varOmega \end{aligned}$$

(5.1.19)

and under the boundary conditions

$$\begin{aligned} \displaystyle \frac{\partial u}{\partial \nu }=\displaystyle \frac{\partial v}{\partial \nu } =\displaystyle \frac{\partial w}{\partial \nu }=0 ~~\hbox {on}~ \partial \varOmega \end{aligned}$$

(5.1.20)

in a bounded convex domain $\varOmega \subset \mathbb {R}^2$ with smooth boundary $\partial \varOmega$.

In what follows, for simplicity, we shall drop the differential element in the integrals without confusion, namely abbreviating $\int _\varOmega f(x) dx$ as $\int _\varOmega f$ and $\int ^t_0\int _\varOmega f(x,\tau ) dxd\tau$ as $\int ^t_0\int _\varOmega f(\cdot ,) d\tau$ as an important step toward a comprehensive understanding of the effect of nonlinear diffusion on the density-suppressed motility model. Our main result asserts that the weak solutions to the density-suppressed motility system (5.1.18) may approach the relevant homogeneous steady state in the large time limit if D is suitably large, which is stated as follows (Xu and Wang 2021).

Theorem 5.3

Let $\varOmega \subset \mathbb {R}^2$ be a bounded convex domain with smooth boundary, and suppose that $m>1,\alpha>0,\beta >0$ and f satisfies (5.1.17). Assume that initial data $(u_0,v_0,w_0)\in (W^{1,\infty }(\varOmega ))^3$ with $u_0\gneqq 0 ,w_0\gneqq 0$ and $v_0>0$ in $\overline{\varOmega }$. Then problem (5.1.18)–(5.1.20) admits at least one global weak solution (u, v, w) in the sense of Definition 2.1 below. Moreover, there exists constant $D_0>0$ such that if $D>D_0$,

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\Vert u(\cdot ,t)-u_{\star }\Vert _{L^{\infty }(\varOmega )}+\Vert v(\cdot ,t)-u_{\star }\Vert _{L^{\infty }(\varOmega )}+\Vert w(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}=0 \end{aligned}$$

(5.1.21)

with $u_{\star }=\frac{1}{|\varOmega |}\int _{\varOmega }u_0+\frac{\beta }{|\varOmega |}\int _{\varOmega }w_0$.

Proof strategies for Theorem 5.3. As the first step to prove the above claim, in Sect. 5.4, we give the definition of a global weak solution to problem (5.1.18)–(5.1.20) and recall that problem (5.1.18)–(5.1.20) with $m>1$ and $\alpha >0$ possesses a globally defined weak solution in two-dimensional setting by the approximation procedure (5.2.20). With respect to the convergence properties asserted in (5.1.21), our analysis is essentially different from that of Jin et al. (2020). In fact, thanks to $\gamma _1\le \gamma (v)\le \gamma _2$ for all $v\ge 0$ in (H), authors of Jin et al. (2020) derived the estimate of $\Vert u(\cdot ,t)\Vert _{L^2(\varOmega )}$, which is the starting point of a priori estimate of $\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}$. In particular, the assumption $\gamma _1\le \gamma (v)$ plays an essential role in constructing energy function $\mathscr {F}(u,v):=\Vert u(\cdot ,t)-u_*\Vert _{L^2(\varOmega )}+ \Vert v(\cdot ,t)-u_*\Vert _{L^2(\varOmega )}$, which leads to the convergence of (u, v) if D is suitable large (see the proofs of Lemma 4.8 and Lemma 4.10 in Jin et al. 2020 for the details). Whereas our asymptotic analysis consists at its core in an analysis of the functional

$$ \int _{\varOmega }u^2+\eta \int _{\varOmega }|\nabla v|^2 $$

for solutions of certain regularized versions of (5.1.18), provided that in dependence on the model parameter D, the positive constant $\eta$ is suitably chosen when D is suitable large. This yields the finiteness of $\int ^{\infty }_{3}\int _{\varOmega }|\nabla u^{\frac{m+1}{2}}|^2$ and $\int ^{\infty }_{3}\int _{\varOmega }|\nabla v |^2$ (see Lemma 5.18) and then entails that as a consequence of these integral inequalities, all our solutions asymptotically become homogeneous in space and hence satisfy (5.1.21) (Lemmas 5.19–5.22).

Remark 5.2

(1) Note that as an apparently inherent drawback, assumption (H) in Jin et al. (2020) excludes $\gamma (v)$ decay functions such as $v^{-\alpha }$. Indeed, despite v is bounded below by $\delta$ with the help of Lemma 5.4 and thereby the upper bound for $\gamma (v)$ can be removed, an lower bound for $\gamma (v)$ in (H) is essentially required therein.

(2) Due to the results on the existence of global solutions in Winkler (2020), the asymptotic behavior of solutions herein seems to be achieved for the higher dimensional version of (5.1.18) at the cost of additional constraint on m and $\alpha$.

5.2 Preliminaries

In this section, we first introduce some notations/definitions and list some basic facts which will be used in our subsequent analysis in Sect. 5.3. In particular, the construction of relaxed super and sub-solutions with spatially inhomogeneous decay rates will be presented. For $c\ge 2\sqrt{a}$, define

$$\begin{aligned} \lambda :=\frac{c-\sqrt{c^2-4a}}{2}\quad \hbox {and}\quad \theta _1(x):=\frac{c-\sqrt{c^2-4a\Big (1+\frac{e^{-\lambda x}}{1+a}\Big )^{-m}}}{2\Big (1+\frac{e^{-\lambda x}}{1+a}\Big )^{-m}}\quad \forall x\in \mathbb R, \end{aligned}$$

(5.2.1)

for which

$$\begin{aligned} \lambda ^2-c\lambda +a=0,\qquad \Big (1+\frac{e^{-\lambda x}}{1+a}\Big )^{-m}\theta _1^2(x)-c\theta _1(x)+a=0\quad \forall x\in \mathbb R \end{aligned}$$

(5.2.2)

and

$$\begin{aligned} \lim _{x\rightarrow +\infty }\theta _1(x)=\lambda ,\quad 0<\theta _1(x)<\lambda \le \sqrt{a}\quad \forall x\in \mathbb R. \end{aligned}$$

(5.2.3)

Choose

$$\begin{aligned} \theta _2(x):=\left\{ \begin{aligned} \theta _1(x)+\lambda /4, \quad c=2\sqrt{a}, \\ \theta _1(x)+\lambda /k_0,\quad c>2\sqrt{a} \end{aligned}\right. \quad \forall x\in \mathbb R \end{aligned}$$

(5.2.4)

with $k_0>\max \left\{ \frac{2\lambda }{c-2\lambda },2\right\}$. Then

$$\begin{aligned} \theta _2(x)\in \left( \theta _1(x),\theta _1(x)+\frac{\lambda }{2}\right) \quad \forall x\in \mathbb R \end{aligned}$$

(5.2.5)

and there exists $x_0\in \mathbb {R}$ such that

$$\begin{aligned} \theta _2(x)<2\theta _1(x)\quad \hbox {for}\; x>x_0. \end{aligned}$$

(5.2.6)

Define two functions:

$$\begin{aligned} \overline{U}(x):=\min \{e^{-\lambda x},\eta \}\quad \forall x\in \mathbb R \end{aligned}$$

(5.2.7)

and

$$\begin{aligned} \underline{U}_n(x):=\left\{ \begin{aligned}&\delta ,&\quad x\le x_\delta , \\&d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x) x},&\quad x>x_\delta \end{aligned} \right. \end{aligned}$$

(5.2.8)

for $b> b^*(m,a)$ with $b^*(m,a)$ is defined in (5.1.12), where $\delta$ is chosen sufficiently small, $x_\delta >0$ is the unique positive solution of the equation $d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x) x}=\delta$,

$$\begin{aligned} d_n:=1-\frac{1}{n}\quad \hbox {with}\; 2\le n\in \mathbb N,\qquad d_0:=\left\{ \begin{aligned} 1,&c=2\sqrt{a}, \\ -1,&c>2\sqrt{a} \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} \eta :=\frac{2a}{b-3m+\sqrt{\left( b-3m\right) ^2-\frac{4m(m+1)a}{1+a}}}. \end{aligned}$$

(5.2.9)

Noticing that $d_n\in (0,1)$ and

$$\lim _{x\rightarrow +\infty }e^{(\theta _1(x)-\lambda )x}=1,$$

which will be verified in Lemma 5.1, we can choose sufficiently small $\delta$, with which $x_\delta$ is large enough such that for all $x\in \mathbb R$,

$$0<\underline{U}_n<\overline{U}\le \eta .$$

We note that the functions $\overline{U}(x)$ and $\underline{U}_n(x)$ will be essentially used later as the super- and sub-solutions of an auxiliary problem we introduce in Sect. 5.3.2. A schematic of $\overline{U}(x)$ and $\underline{U}_n(x)$ is plotted in Fig. 5.2. Note that the coefficients $d_n$ ($n\ge 2$) and $d_0$ determine the amplitude of $\underline{U}_n(x)$ and $\theta _1(x)$ and determine the decay of $\underline{U}_n(x)$ for large $x>x_\delta$. This is a new ingredient developed in the first part of Chapter 5 to settle the difficulty of analysis caused by the nonlinear motility function $\gamma (v)$.

Fig. 5.2

A schematic of functions $\overline{U}(x)$ and $\underline{U}_n(x)$, where the solid black line represents $\overline{U}(x)$ and the dashed red line represents $\underline{U}_n(x)$

Denote

$$C_{\mathrm {unif}}^b(\mathbb R):=\{u\in C(\mathbb R)|\ u\;\hbox {is uniformly continuous in}\; \mathbb R \ \mathrm {and} \ \sup _{z\in \mathbb R}|u(z)|<+\infty \},$$

which is equipped with the norm

$$\Vert u\Vert =\sup _{z\in \mathbb R}|u(z)|.$$

Define the function space

$$\begin{aligned} \mathscr {E}_n:=\{u\in C_{\mathrm {unif}}^b(\mathbb R)|\underline{U}_n\le u\le \overline{U}\}, \ \ \mathscr {X}_0:=\bigcap _{n>1}\mathscr {E}_n. \end{aligned}$$

To find solutions of (5.1.10) in $\mathscr {X}_0$, we need the following Lemmas.

Lemma 5.1

Let $\lambda$ and $\theta _1(x)$ be defined in (5.2.1). Then it follows that

$$\begin{aligned} \lim _{x\rightarrow +\infty }e^{(\theta _1(x)-\lambda )x}=1. \end{aligned}$$

(5.2.10)

Moreover, for sufficiently small $\delta >0$, if $x>x_\delta$, then for $c=2\sqrt{a}$,

$$\begin{aligned} 0<\theta _1'(x)\le 2K_1e^{-\frac{\lambda }{2} x} \ \mathrm {and}\ -\lambda K_1e^{-\frac{\lambda }{2} x}\le \theta _1''(x)<0 \end{aligned}$$

(5.2.11)

with $K_1=\frac{a}{2}\sqrt{\frac{m}{1+a}}$; while for $c>2\sqrt{a}$,

$$\begin{aligned} 0<\theta _1'(x)\le 2K_2e^{-\lambda x} \ \mathrm {and} \ -2\lambda K_2e^{-\lambda x}\le \theta _1''(x)<0 \end{aligned}$$

(5.2.12)

with $K_2=\frac{4a^2m\lambda }{\left( c+\sqrt{c^2-4a}\right) ^2\sqrt{c^2-4a}(1+a)}.$

Proof

In the sequel, for notational simplicity, under $c\ge 2\sqrt{a}$, we introduce the following notations:

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi (x):=\displaystyle 1+\frac{e^{-\lambda x}}{1+a},\\ \rho (\phi (x)):=\sqrt{c^2-4a\phi ^{-m}(x)},\\ h(\phi (x)):=\displaystyle \frac{2a}{c+\rho (\phi (x))}=\displaystyle \frac{2a}{c+\sqrt{c^2-4a\phi ^{-m}(x)}}. \end{array}\right. } \end{aligned}$$

Then from the definition of $\theta _1(x)$, we have

$$\theta _1(x)=\frac{c-\sqrt{c^2-4a\phi ^{-m}(x)}}{2\phi ^{-m}(x)}=\frac{2a}{c+\sqrt{c^2-4a\phi ^{-m}(x)}}=h(\phi (x)).$$

With simple calculation, we find

$$\begin{aligned} h'(\phi (x))=\frac{-4a^2m}{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi ^{m+1}(x)},\quad \phi '(x)=\frac{-\lambda e^{-\lambda x}}{1+a} \end{aligned}$$

(5.2.13)

and then

$$\begin{aligned} \theta _1'(x)&=(h(\phi (x)))'=h'(\phi )\phi '(x)=\frac{4a^2m\lambda e^{-\lambda x}}{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi ^{m+1}(x)(1+a)}>0. \end{aligned}$$

(5.2.14)

When $c=2\sqrt{a}$, it has that $\lim \limits _{x\rightarrow +\infty }\phi (x)=1$ and $\lim \limits _{x\rightarrow +\infty }\rho (\phi (x))=0$. By L’Hopital’s rule, we have

$$\begin{aligned} \lim _{x\rightarrow +\infty }\bigg |\frac{e^{-\frac{\lambda }{2} x}}{\rho (\phi (x))}\bigg |^2&=\lim _{x\rightarrow +\infty }\frac{e^{-\lambda x}}{c^2-4a\phi ^{-m}(x)}\nonumber \\&=\lim _{x\rightarrow +\infty }\frac{e^{-\lambda x}}{4a\phi ^m(x)-4a}\lim _{x\rightarrow +\infty }\phi ^m(x) \\&=\lim _{x\rightarrow +\infty }\frac{1+a}{4ma\phi ^{m-1}(x)}=\frac{1+a}{4ma}.\nonumber \end{aligned}$$

(5.2.15)

Then it can be easily verified that

$$\lim _{x\rightarrow +\infty }\frac{4a^2m\lambda e^{-\frac{\lambda }{2}x}}{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi ^{m+1}(x)(1+a)}=\frac{a}{2}\sqrt{\frac{m}{1+a}}:=K_1,$$

from which and (5.2.14), by choosing sufficiently small $\delta >0$, we can find $x_\delta >0$ such that for $x>x_\delta$, there holds that

$$0<\theta _1'(x)\le 2K_1e^{-\frac{\lambda }{2}x}.$$

When $c>2\sqrt{a}$, it has that $\lim \limits _{x\rightarrow +\infty }\phi (x)=1$ and $\lim \limits _{x\rightarrow +\infty }\rho (\phi (x))=\sqrt{c^2-4a}$. It can be directly checked that

$$\begin{aligned}&\lim _{x\rightarrow +\infty }\frac{4a^2m\lambda }{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi ^{m+1}(x)(1+a)} \\ =&\,\,\frac{4a^2m\lambda }{\sqrt{c^2-4a}(c+\sqrt{c^2-4a})^2(1+a)}:=K_2. \end{aligned}$$

Then from (5.2.14), by choosing sufficiently small $\delta >0$, for $x>x_\delta$, we obtain

$$0<\theta _1'(x)\le 2K_2e^{-\lambda x}.$$

The first parts of (5.2.11) and (5.2.12) are proved.

On the other hand, by L’Hôpital’s rule, using (5.2.13) and (5.2.15), for $c=2\sqrt{a}$, we obtain

$$\begin{aligned}&\lim _{x\rightarrow +\infty }\left( h(\phi (x))-h(1)\right) x \\ =&\,\,\lim _{x\rightarrow +\infty }h'(\phi (x))\frac{\lambda x^2e^{-\lambda x}}{1+a} \\ =&\,\,\lim _{x\rightarrow +\infty }\frac{-4a^2m\lambda x^2 e^{-\lambda x}}{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi ^{m+1}(x)(1+a)} \\ =&\,\,-\frac{4a^2m\lambda }{c^2(1+a)}\lim _{x\rightarrow +\infty }\frac{e^{-\frac{\lambda }{2} x}}{\rho (\phi (x))}\lim _{x\rightarrow +\infty }\frac{x^2}{e^{\frac{\lambda }{2} x}} \\ =&\,\,-\frac{am\lambda }{1+a}\sqrt{\frac{1+a}{4ma}}\lim _{x\rightarrow +\infty }\frac{x^2}{e^{\frac{\lambda }{2} x}}=0. \end{aligned}$$

While for $c>2\sqrt{a}$, we obtain

$$\begin{aligned} \lim _{x\rightarrow +\infty }\left( h(\phi (x))-h(1)\right) x=&\,\,\lim _{x\rightarrow +\infty }h'(\phi (x))\frac{\lambda x^2e^{-\lambda x}}{1+a} =\frac{\lambda h'(1)}{1+a}\lim _{x\rightarrow +\infty }\frac{x^2}{e^{\lambda x}}=0. \end{aligned}$$

Summing up, for $c\ge 2\sqrt{a}$, we obtain

$$\lim _{x\rightarrow +\infty }e^{(\theta _1(x)-\lambda )x}=e^{\lim \limits _{x\rightarrow +\infty }(h(\phi (x))-h(1))x}=1$$

and then (5.2.10) follows.

Now, we turn to the estimate of $\theta _1''(x)$. Noticing that $h''(\phi )=h'(\phi )(\ln (- h'(\phi )))'$ and

$$\begin{aligned} (\ln (-h'(\phi )))'=&\,\,\left[ \ln (4a^2m)-2\ln (c+\rho (\phi ))-\ln \rho (\phi )-(m+1)\ln \phi \right] '\nonumber \\ =&\,\,\frac{-4am}{\rho (\phi )(c+\rho (\phi ))\phi ^{m+1}}-\frac{2am}{(\rho (\phi ))^2\phi ^{m+1}}-\frac{m+1}{\phi },\nonumber \end{aligned}$$

which together with (5.2.13) and the fact that $\phi '(x)=\frac{-\lambda e^{-\lambda x}}{1+a}$ and $\phi ''(x)=\frac{\lambda ^2 e^{-\lambda x}}{1+a}$ implies

$$\begin{aligned}&\quad \theta _1''(x)\\&=(h(\phi (x))''=(h'(\phi (x))\phi '(x))'=h'(\phi (x))\phi ''(x)+h''(\phi (x))(\phi '(x))^2 \\&=h'(\phi (x))[\phi ''(x)+(\ln (-h'(\phi )))'(\phi '(x))^2] \\&=\frac{-4a^2m\lambda ^2e^{-\lambda x}}{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi (x)^{m+1}(1+a)} \\&\quad \cdot \left\{ 1-\frac{e^{-\lambda x}}{1+a}\left( \frac{4am}{\rho (\phi (x))(c+\rho (\phi (x)))\phi (x)^{m+1}}+\frac{2am}{(\rho (\phi (x)))^2\phi (x)^{m+1}}+\frac{m+1}{\phi (x)}\right) \right\} . \end{aligned}$$

For $c=2\sqrt{a}$, then $\lambda =\sqrt{a}$, from (5.2.15), it can be verified that

$$\lim _{x\rightarrow +\infty }\frac{-4a^2m\lambda ^2e^{-\frac{\lambda }{2} x}}{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi (x)^{m+1}(1+a)}=-\lambda K_1$$

and

$$\lim _{x\rightarrow +\infty }\frac{e^{-\lambda x}}{1+a}\left( \frac{4am}{\rho (\phi (x))(c+\rho (\phi (x)))\phi (x)^{m+1}} +\frac{2am}{(\rho (\phi (x)))^2\phi (x)^{m+1}}+\frac{m+1}{\phi (x)}\right) =\frac{1}{2}.$$

By choosing sufficiently small $\delta >0$, we can find a $x_\delta >0$ such that

$$0>\theta _1''(x)\ge -\lambda K_1e^{-\frac{\lambda }{2}x}, \ \mathrm {for} \ x>x_\delta .$$

While for $c>2\sqrt{a}$, we can check that

$$\lim _{x\rightarrow +\infty }\frac{-4a^2m\lambda ^2}{\rho (\phi (x))(c+\rho (\phi (x)))^2\phi (x)^{m+1}(1+a)}=-\lambda K_2$$

and

$$\lim _{x\rightarrow +\infty }\frac{e^{-\lambda x}}{1+a}\left( \frac{4am}{\rho (\phi (x))(c+\rho (\phi (x)))\phi (x)^{m+1}} +\frac{2am}{(\rho (\phi (x)))^2\phi (x)^{m+1}}+\frac{m+1}{\phi (x)}\right) =0.$$

Then by choosing sufficiently small $\delta >0$ so as to generate a $x_\delta >0$, we have

$$\begin{aligned} 0>\theta _1''(x)\ge -2\lambda K_2e^{-\lambda x}, \ \mathrm {for} \ x>x_\delta . \end{aligned}$$

Then the last parts of (5.2.11) and (5.2.12) follow. This completes the proof of Lemma 5.1.

Throughout Sect. 5.4, we shall pursue weak solutions to problem (5.1.18)–(5.1.20) specified as follows.

Definition 5.1

Let $m>1,\alpha>0,\beta >0$ and f satisfies (5.1.17). Then a triple (u, v, w) of nonnegative functions

$$ \left\{ \begin{aligned} u\in L^1_{loc}(\overline{\varOmega }\times [0,\infty ))\\ v\in L^1_{loc}([0,\infty );W^{1,1}(\varOmega ))\\ w\in L^1_{loc}([0,\infty );W^{1,1}(\varOmega )) \end{aligned} \right. $$

will be called a global weak solution of problem (5.1.18)–(5.1.20) if

$$\begin{aligned} u^m /v^\alpha \in L^1_{loc}(\overline{\varOmega }\times [0,\infty )) \end{aligned}$$

(5.2.16)

and

$$\begin{aligned} -\int ^{\infty }_0\int _{\varOmega }u\varphi _t-\int _{\varOmega }u_0\varphi (\cdot ,0)=\int ^{\infty }_0\int _{\varOmega }\frac{u^m}{v^\alpha }\varDelta \varphi + \beta \int ^{\infty }_0\int _{\varOmega }uf(w)\varphi \end{aligned}$$

(5.2.17)

for all $\varphi \in C^{\infty }_0(\overline{\varOmega }\times [0,\infty ))$ such that $\frac{\partial \varphi }{\partial \nu }|_{\partial \varOmega }=0$ and

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

(5.2.18)

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

$$\begin{aligned} \int ^{\infty }_0\int _{\varOmega } w\varphi _t-\int _{\varOmega }w_0\varphi (\cdot ,0)=-\int ^{\infty }_0\int _{\varOmega }\nabla w\cdot \nabla \varphi -\int ^{\infty }_0\int _{\varOmega }uf(w)\varphi \end{aligned}$$

(5.2.19)

for all $\varphi \in C^{\infty }_0(\overline{\varOmega }\times [0,\infty ))$.

For $\varepsilon \in (0,1)$, we denote by $(u_{\varepsilon },v_{\varepsilon },w_{\varepsilon })$ the solution of the regularized problem

$$\begin{aligned} \left\{ \begin{aligned}&u_{\varepsilon t}=\varepsilon \varDelta (u_{\varepsilon }+1)^{M}+\varDelta \left( u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}v_{\varepsilon }^{-\alpha }\right) +\beta u_{\varepsilon }f(w_{\varepsilon }),&x\in \varOmega ,~t>0, \\&v_{\varepsilon t}=D\varDelta v_{\varepsilon }+u_{\varepsilon }-v_{\varepsilon },&x\in \varOmega ,~t>0,\\&w_{\varepsilon t}=\varDelta w_{\varepsilon }-u_{\varepsilon }f(w_{\varepsilon }),&x\in \varOmega ,~t>0, \\&\frac{\partial u_{\varepsilon }}{\partial \nu }=\frac{\partial v_{\varepsilon }}{\partial \nu }=\frac{\partial w_{\varepsilon }}{\partial \nu }=0,&x\in \partial \varOmega ,~t>0,\\&u_{\varepsilon }(x,0)=u_0,~v_{\varepsilon }(x,0)=v_0,~w_{\varepsilon }(x,0)=w_0,&x\in \varOmega \end{aligned} \right. \end{aligned}$$

(5.2.20)

with $M>m$. Note that due to the a priori boundedness of $w_{\varepsilon }$, the global smooth solvability of (5.2.20) can be derived by the argument in Lemma 2.4 of Winkler (2020) with evident minor adaptations, and we may refrain from giving the details for brevity here. As for the global weak solutions of (5.1.18)–(5.1.20), we can state as follows.

Lemma 5.2

Let $m>1,\alpha>0,\beta >0$ and f satisfies (5.1.17). Then there exist $(\varepsilon _{j})_{j\in \mathbb {N}}\subset (0,1)$ as well as nonnegative functions

$$\begin{aligned} \left\{ \begin{aligned}&u\in L^{\infty }(\overline{\varOmega }\times [0,\infty )) \\&v\in C^0(\overline{\varOmega }\times [0,\infty ))\bigcap L^2_{loc}([0,\infty );W^{1,2}(\varOmega ))\\&w\in C^0(\overline{\varOmega }\times [0,\infty ))\bigcap L^2_{loc}([0,\infty );W^{1,2}(\varOmega )) \end{aligned} \right. \end{aligned}$$

(5.2.21)

such that $\varepsilon _{j}\searrow 0$ as $j\rightarrow \infty$ and as $\varepsilon _{j}\searrow 0$, we have

$$\begin{aligned}&u_{\varepsilon }\rightarrow u\quad \text {a.e. in }\varOmega \times (0,\infty ),\end{aligned}$$

(5.2.22)

$$\begin{aligned}&u_{\varepsilon }\rightarrow u\quad \text {in }\bigcap _{p\ge 1}L^p_{loc}(\overline{\varOmega }\times [0,\infty )),\end{aligned}$$

(5.2.23)

$$\begin{aligned}&v_{\varepsilon }\rightarrow v\quad \text {in }C_{loc}^0(\overline{\varOmega }\times [0,\infty )),\end{aligned}$$

(5.2.24)

$$\begin{aligned}&w_{\varepsilon }\rightarrow w\quad \text {in }C_{loc}^0(\overline{\varOmega }\times [0,\infty )),\end{aligned}$$

(5.2.25)

$$\begin{aligned}&\nabla v_{\varepsilon }\rightharpoonup \nabla v \quad \text {in }L^2_{loc}(\overline{\varOmega }\times [0,\infty )),\end{aligned}$$

(5.2.26)

$$\begin{aligned}&\nabla w_{\varepsilon }\rightharpoonup \nabla w \quad \text {in }L^2_{loc}(\overline{\varOmega }\times [0,\infty )). \end{aligned}$$

(5.2.27)

Moreover, $v>0$ in $\overline{\varOmega }\times (0,\infty )$ and (u, v, w) forms a global weak solution of (5.1.18)–(5.1.20) in the sense of Definition 5.1.

Proof

The existence of global weak solutions of (5.1.18)–(5.1.20) can be verified on the basis of straightforward extraction procedures as in Winkler (2020). Indeed, due to the a priori boundedness of $w_{\varepsilon }$, one can derive some necessary a priori estimation for $(u_{\varepsilon },v_{\varepsilon },w_{\varepsilon })$ such as $\int ^{t+1}_t \int _{\varOmega } u_\varepsilon ^p$ with all $p<m+1$, $(v_\varepsilon ,w_\varepsilon )$ in $(W^{1,q}(\varOmega ))^2$ with some $q>2$ and $u_\varepsilon$ in $L^\infty (\varOmega )$ and finally apply an Aubin–Lions lemma to obtain a weak solution of (5.1.18)–(5.1.20) with the additional information (5.2.22) (we refer the reader to the proof of Lemma 7.1 in Winkler 2020 for detail).

The following basic properties of the spatial $L^1$ norms of $(u_\varepsilon ,v_\varepsilon ,w_\varepsilon )$ as well as the $L^\infty$ norm of $w_\varepsilon$ are easily verified.

Lemma 5.3

Let $(u_{\varepsilon },v_{\varepsilon },w_{\varepsilon })$ be the classical solution of (5.2.20) in $\varOmega \times (0,\infty )$. Then we have

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^1(\varOmega )}+\beta \Vert w_{\varepsilon }(\cdot ,t)\Vert _{L^1(\varOmega )}=\Vert u_0\Vert _{L^1(\varOmega )}+\beta \Vert w_0\Vert _{L^1(\varOmega )}, \end{aligned}$$

(5.2.28)

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^1(\varOmega )} \ge \Vert u_0\Vert _{L^1(\varOmega )}, \end{aligned}$$

(5.2.29)

$$\begin{aligned} \int _{\varOmega }v_{\varepsilon }(\cdot ,t)\le \int _{\varOmega } v_0+\int _{\varOmega }u_0+\beta \int _{\varOmega }w_0 \end{aligned}$$

(5.2.30)

as well as

$$\begin{aligned} t\mapsto \Vert w_{\varepsilon }(\cdot ,t)\Vert _{L^{\infty }(\varOmega )} ~\text {is non-increasing in }~~[0,\infty ). \end{aligned}$$

(5.2.31)

Proof

Multiplying $w_{\varepsilon }$-equation by $\beta$ and adding the result to $u_{\varepsilon }$-equation in (5.2.20), we get

$$\begin{aligned} \beta \frac{d}{dt}\int _{\varOmega }w_{\varepsilon }+\frac{d}{dt}\int _{\varOmega }u_{\varepsilon }=0, \end{aligned}$$

(5.2.32)

which immediately yields (5.2.28). An integration of the first equation in (5.2.20) gives us

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }u_{\varepsilon }=\int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon })\ge 0 \end{aligned}$$

(5.2.33)

which readily entails (5.2.29). Upon the integration of the second equation in (5.2.20), we can see that

$$ \frac{d}{dt}\int _{\varOmega }v_{\varepsilon }+\int _{\varOmega }v_{\varepsilon }\le \int _{\varOmega }u_{\varepsilon } $$

which, along with (5.2.28), leads to (5.2.30). Due to the fact that f and $w_{\varepsilon }$ are nonnegative, the claim in (5.2.31) results upon an application of the maximum principle to $w_{\varepsilon }$-equation in (5.2.20).

Let us first derive a positive uniform-in-time lower bound for $v_\varepsilon$ which will alleviate the difficulties caused by the singularity of signal-dependent motility function $v^{-\alpha }$ near zero. Despite the quantitative lower estimate for solutions of the Neumann problem was established in the related literature (Hillen et al. 2013; Winkler 2020), we present a proof of our results with some necessary details to make the lower bound accessible to the sequel analysis.

Lemma 5.4

For all $D\ge 1$ and $\varepsilon \in (0,1)$, there exists $\delta >0$ such that

$$\begin{aligned} v_{\varepsilon }(x,t)>\delta ~\quad \hbox {for all} ~x\in \varOmega ~ \hbox {and}~ t>0. \end{aligned}$$

(5.2.34)

Proof

According to the pointwise lower bound estimate for the Neumann heat semigroup $(e^{t\varDelta })_{t\ge 0}$ on the convex domain $\varOmega$, one can find $C_1(\varOmega )>0$ such that

$$ e^{t\varDelta }\varphi \ge C_1(\varOmega )\int _{\varOmega }\varphi \qquad \text {for all }t\ge 1\text { and each nonnegative }\varphi \in C^0(\overline{\varOmega }) $$

(e.g., Fujie 2016; Hillen et al. 2013).

By the time rescaling $\tilde{t}=Dt$, we can see that $\tilde{v}(x,\tilde{t}):=v_{\varepsilon }(x,\frac{\tilde{t}}{D})$ satisfies

$$\begin{aligned} \frac{\partial \tilde{v}}{\partial \tilde{t}}=\varDelta \tilde{v}-D^{-1}\tilde{v}+D^{-1} u_{\varepsilon }(x,D^{-1}\tilde{t}). \end{aligned}$$

(5.2.35)

Now applying the variation-of-constants formula to (5.2.35), we have

$$\begin{aligned} \begin{aligned} \tilde{v} (\cdot ,\tilde{t})&=e^{\tilde{t}(\varDelta -D^{-1})}v_0(\cdot )+ D^{-1}\displaystyle \int ^{\tilde{t}}_{0}e^{(\tilde{t}-s)(\varDelta -D^{-1})}u_{\varepsilon }(\cdot ,D^{-1}s)ds\quad ~t>0, \end{aligned} \end{aligned}$$

(5.2.36)

where by the comparison principle, we can see

$$\begin{aligned} \begin{aligned} e^{\tilde{t}(\varDelta -D^{-1})} v_0(\cdot )&\ge e^{-\tilde{t} D^{-1}}\displaystyle \inf _{x\in \varOmega }v_0(x)\\&\ge e^{- 2 }\displaystyle \inf _{x\in \varOmega }v_0(x)\quad \hbox {for all}~~x\in \varOmega , \tilde{t}\le 2D \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\quad D^{-1}\displaystyle \int ^{\tilde{t}}_{0}e^{(\tilde{t}-s)(\varDelta -D^{-1})}u_{\varepsilon }(\cdot ,D^{-1}s)ds\\&\ge D^{-1}\displaystyle \int ^{\tilde{t}-1}_{0}e^{(\tilde{t}-s)(\varDelta -D^{-1})}u_{\varepsilon }(\cdot ,D^{-1}s)ds\\&\ge C_1(\varOmega ) D^{-1}(\displaystyle \int ^{\tilde{t}-1}_0 e^{-D^{-1}(\tilde{t}-s)}ds) \inf \limits _{s\in (0,\infty )}\int _{\varOmega }u_{\varepsilon }(\cdot ,s)\\&\ge C_1(\varOmega ) (e^{-D^{-1}}-e^{-D^{-1}\tilde{t}})\displaystyle \int _{\varOmega }u_{0}\\&\ge \displaystyle \frac{C_1(\varOmega )}{2e}\int _{\varOmega }u_{0}\quad \hbox {for all}~~x\in \varOmega ~\hbox {and}~\tilde{t}\ge 2D, \end{aligned} \end{aligned}$$

due to $D\ge 1$. Therefore, inserting above inequalities into (5.2.36), readily establish (5.2.34) with $\delta = \min \{\frac{C_1(\varOmega )}{2e}\int _{\varOmega }u_{0}, e^{- 2 }\displaystyle \inf _{x\in \varOmega }v_0(x)\}$.

Through a straightforward semigroup argument, we formulate a favorable dependence of $\Vert v_{\varepsilon }(\cdot ,t)\Vert _{L^{p}(\varOmega )}$ with respect to parameter D.

Lemma 5.5

For $p>1$, there exists $C(p)>0$ such that

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

(5.2.37)

Proof

Applying Duhamel’s formula to the equation

$$\begin{aligned} \frac{\partial \tilde{v}}{\partial \tilde{t}}=\varDelta \tilde{v}-D^{-1}\tilde{v}+D^{-1} u_{\varepsilon }(x,D^{-1}\tilde{t}) \end{aligned}$$

satisfied by $\tilde{v}(x,\tilde{t}):=v_{\varepsilon }(x,\frac{\tilde{t}}{D})$ and employing well-known smoothing properties of the Neumann heat semigroup $(e^{t\varDelta })_{t\ge 0}$ on $\varOmega$ (see Lemma 3 of Rothe 1984 or Lemma 1.3 of Winkler 2010 for example), we can find $C_p>0$ such that for any $\tilde{t}>0$

$$\begin{aligned} \begin{aligned}&\Vert \tilde{v}_{\varepsilon }(\cdot ,\tilde{t})\Vert _{L^p(\varOmega )}\\ =&\,\,\Vert e^{-D^{-1}\tilde{t}} e^{\tilde{t} \varDelta } v_0(\cdot )+D^{-1}\displaystyle \int ^{\tilde{t}}_{ 0}e^{(\tilde{t}-s)(\varDelta -D^{-1}) }u_{\varepsilon }(\cdot ,D^{-1}s)ds\Vert _{L^p(\varOmega )}\\ \le&\,\, e^{-D^{-1} \tilde{t}}\Vert v_0\Vert _{L^p(\varOmega )} +\displaystyle \frac{C_p}{D}\displaystyle \int ^{\tilde{t}}_{ 0}e^{-D^{-1}(\tilde{t}-s) }(1+(\tilde{t}-s)^{-1+\frac{1}{p}})\Vert u_{\varepsilon }(\cdot ,D^{-1}s)\Vert _{L^1(\varOmega )}ds\\ \le&\,\, \Vert v_0\Vert _{L^p(\varOmega )} +\displaystyle \frac{C_p}{D}(\Vert u_0\Vert _{L^1(\varOmega )}+\beta \Vert w_0\Vert _{L^1(\varOmega )}) \displaystyle \int ^{\tilde{t}}_{ 0}e^{-D^{-1}(\tilde{t}-s) }(1+(\tilde{t}-s)^{-1+\frac{1}{p}}) ds\\ =&\,\, \Vert v_0\Vert _{L^p(\varOmega )} +\displaystyle \frac{C_p}{D}(\Vert u_0\Vert _{L^1(\varOmega )}+\beta \Vert w_0\Vert _{L^1(\varOmega )}) \displaystyle \int ^{\tilde{t}}_{ 0}e^{-D^{-1}\sigma }(1+\sigma ^{-1+\frac{1}{p}}) d\sigma \\ \le&\,\, \Vert v_0\Vert _{L^p(\varOmega )} +\displaystyle \frac{C_p}{D}(\Vert u_0\Vert _{L^1(\varOmega )}+\beta \Vert w_0\Vert _{L^1(\varOmega )})(D+D^{\frac{1}{p}} \displaystyle \int ^{\infty }_{0}e^{-\sigma }\sigma ^{-1+\frac{1}{p}} d\sigma )\\ \le&\,\, \Vert v_0\Vert _{L^p(\varOmega )} + (1+D^{\frac{1}{p}-1})C_p (\Vert u_0\Vert _{L^1(\varOmega )}+\beta \Vert w_0\Vert _{L^1(\varOmega )})(1+\displaystyle \int ^{\infty }_{0}e^{-\sigma }\sigma ^{-1+\frac{1}{p}} d\sigma ) \end{aligned} \end{aligned}$$

which ends up (5.2.37) with

$$C(p)= \Vert v_0\Vert _{L^p(\varOmega )}+2 C_p (\Vert u_0\Vert _{L^1(\varOmega )}+\beta \Vert w_0\Vert _{L^1(\varOmega )}) \left( 1+\displaystyle \int ^{\infty }_{0}e^{-\sigma }\sigma ^{-1+\frac{1}{p}} d\sigma \right) .$$

5.3 Traveling Wave Solutions to a Density-Suppressed Motility Model

5.3.1 Some a Priori Estimates

Lemma 5.6

For any $u\in \mathscr {E}_n$, denote $V(\cdot ;u)$ the solution of

$$\begin{aligned} V''+cV'+u-V=0. \end{aligned}$$

(5.3.1)

Then for $c\ge 2\sqrt{a}$, we have

$$\begin{aligned}&0<V(x;u)\le \min \left\{ \frac{e^{-\lambda x}}{1+a},\eta \right\} , \end{aligned}$$

(5.3.2)

$$\begin{aligned}&|V'(x;u)| \le \min \left\{ \frac{2\eta }{\sqrt{c^2+4}},\frac{2e^{-\lambda x}}{\sqrt{c^2+4}}\right\} \le \min \left\{ \frac{\eta }{\sqrt{1+a}},\frac{e^{-\lambda x}}{\sqrt{1+a}}\right\} \end{aligned}$$

(5.3.3)

for all $x\in \mathbb R$.

Proof

Denote

$$\begin{aligned} \lambda _1=\frac{-c-\sqrt{c^2+4}}{2},\quad \lambda _2=\frac{-c+\sqrt{c^2+4}}{2}. \end{aligned}$$

(5.3.4)

From (5.3.4) and the definition of $\lambda$ in (5.2.1), we obtain

$$\begin{aligned} 0<\lambda \le \sqrt{a},\quad \lambda _1<0,\quad \lambda _2>0,\quad \lambda _1+\lambda <0,\quad \lambda _2+\lambda >0 \end{aligned}$$

(5.3.5)

and

$$\begin{aligned} \lambda _1\lambda _2=-1,\quad \lambda _1+\lambda _2=-c,\quad \lambda ^2-c\lambda -1=-(1+a). \end{aligned}$$

(5.3.6)

By the variation of constants, the solution of (5.3.1) can be expressed as

$$\begin{aligned} V(x;u)=\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^xe^{\lambda _1(x-s)}u(s)ds+\int _x^{+\infty } e^{\lambda _2(x-s)}u(s)ds\right) . \end{aligned}$$

(5.3.7)

Note that $0\le u\le \overline{U}=\min \{\eta ,e^{-\lambda x}\}$ since $u\in \mathscr {E}_n$. Then using (5.3.5) and (5.3.6), we obtain from (5.3.7) that

$$\begin{aligned} 0\le V(x;u)\le&\,\,\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^xe^{\lambda _1(x-s)}e^{-\lambda s}ds+\int _x^{+\infty } e^{\lambda _2(x-s)}e^{-\lambda s}ds\right) \\ =&\,\,\frac{1}{\lambda _2-\lambda _1}\left( \frac{e^{-(\lambda _1+\lambda )s}\big |_{-\infty }^{x}}{-(\lambda _1+\lambda )e^{-\lambda _1x}} +\frac{e^{-(\lambda _2+\lambda )s}\big |_{x}^{+\infty }}{-(\lambda _2+\lambda )e^{-\lambda _2x}}\right) \\ =&\,\,\frac{-e^{-\lambda x}}{\lambda ^2-c\lambda -1}=\frac{e^{-\lambda x}}{1+a} \end{aligned}$$

and

$$\begin{aligned} 0\le V(x;u)\le&\,\,\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^xe^{\lambda _1(x-s)}\eta ds+\int _x^{+\infty } e^{\lambda _2(x-s)}\eta ds\right) =\frac{-\eta }{\lambda _1\lambda _2} =\eta . \end{aligned}$$

Thus, the inequality in (5.3.2) follows. On the other hand, differentiating (5.3.7) with respect to x, we have

$$\begin{aligned} V'(x;u)=\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^x\lambda _1e^{\lambda _1(x-s)}u(s)ds+\int _x^{+\infty }\lambda _2 e^{\lambda _2(x-s)}u(s)ds\right) . \end{aligned}$$

For $c\ge 2\sqrt{a}$, using (5.3.5), (5.3.6) and the fact that

$$ \lambda _2-\lambda _1=\sqrt{c^2+4}\ge 2\sqrt{1+a}, $$

as well as the fact $0\le u\le \min \{\eta ,e^{-\lambda x}\}$, we obtain with some simple calculations

$$\begin{aligned} |V'(x;u)|\le&\,\,\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^x(-\lambda _1)e^{\lambda _1(x-s)}e^{-\lambda s}ds+\int _x^{+\infty }\lambda _2 e^{\lambda _2(x-s)}e^{-\lambda s}ds\right) \\ \le&\,\,\frac{2e^{-\lambda x}}{\sqrt{c^2+4}}\le \frac{e^{-\lambda x}}{\sqrt{1+a}} \end{aligned}$$

and

$$\begin{aligned} |V'(x;u)|\le&\,\,\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^x(-\lambda _1)e^{\lambda _1(x-s)}\eta ds+ \int _x^{+\infty }\lambda _2e^{\lambda _2(x-s)}\eta ds\right) \\ \le&\,\,\frac{2\eta }{\sqrt{c^2+4}}\le \frac{\eta }{\sqrt{1+a}}, \end{aligned}$$

from which the inequality in (5.3.3) follows. The Lemma is, thus, proved.

Lemma 5.7

For any $u\in \mathscr {E}_n$, denote V(x; u) the solution of

$$\begin{aligned} V''+cV'+u-V=0. \end{aligned}$$

Then for sufficiently small $\delta >0$, if $x>x_\delta$, then

$$\begin{aligned} \gamma (V)\theta _1^2(x)-c\theta _1(x)+a\ge 0, \end{aligned}$$

(5.3.8)

and

$$\begin{aligned}&\gamma (V)\theta _2^2(x)-c\theta _2(x)+a\ge \frac{a}{64} \quad \mathrm {if}\; c=2\sqrt{a}, \end{aligned}$$

(5.3.9)

$$\begin{aligned}&\gamma (V)\theta _2^2(x)-c\theta _2(x)+a\le -\frac{\lambda (c-2\lambda )}{4k_0}\quad \mathrm {if}\; c>2\sqrt{a}. \end{aligned}$$

(5.3.10)

Proof

Noticing $V(x)\le \frac{e^{-\lambda x}}{1+a}$ for $x>x_\delta$, we get (5.3.8) from the fact that

$$\begin{aligned} \gamma (V)\theta _1^2(x)-c\theta _1(x)+a\ge \Big (1+\frac{e^{-\lambda x}}{1+a}\Big )^{-m}\theta _1^2(x)-c\theta _1(x)+a=0. \end{aligned}$$

With

$$\begin{aligned} \lim _{x\rightarrow +\infty }\Big (1+\frac{e^{-\lambda x}}{1+a}\Big )^{-m}=1,\quad \lim _{x\rightarrow +\infty }\theta _1(x)=\lambda , \end{aligned}$$

by choosing sufficiently small $\delta >0$, for all $x>x_\delta$, we have

$$\begin{aligned} \frac{15}{16}\lambda \le \theta _1(x)<\lambda ,\quad \gamma (V)=\frac{1}{(1+V)^m}\ge \Big (1+\frac{e^{-\lambda x}}{1+a}\Big )^{-m}\ge \frac{33}{34}. \end{aligned}$$

(5.3.11)

For the case $c=2\sqrt{a}$, for which $\lambda =\frac{c}{2}$, noticing $\theta _2(x)=\theta _1(x)+\frac{1}{4}\lambda$, using (5.3.11), we get

$$\begin{aligned}&\quad \gamma (V)\theta _2^2(x)-c\theta _2(x)+a\\&=\gamma (V)\theta _1^2(x)-c\theta _1(x)+a+\gamma (V)\left( \frac{1}{16}\lambda ^2+\frac{1}{2}\lambda \theta _1(x)\right) -\frac{1}{4}c\lambda \\&\ge \gamma (V)\left( \frac{1}{16}\lambda ^2+\frac{1}{2}\lambda \theta _1(x)\right) -\frac{1}{4}c\lambda \\&\ge \frac{33}{34}\left( \frac{1}{16}\lambda ^2 +\frac{15}{32}\lambda ^2\right) -\frac{1}{2}\lambda ^2=\frac{a}{64} \end{aligned}$$

for all $x>x_\delta$, from which (5.3.9) follows.

On the other hand, for the case $c>2\sqrt{a}$, for which $\lambda <\frac{c}{2}$, noticing

$$\theta _2(x)=\theta _1(x)+\frac{1}{k_0}\lambda \quad \hbox {with}\;k_0>\max \left\{ \frac{2\lambda }{c-2\lambda },2\right\} ,$$

we obtain

$$\frac{1}{k_0}\lambda ^2+2\lambda \theta _1(x)-c\lambda \le \frac{1}{2}\lambda (2\lambda -c)<0$$

and then

$$\begin{aligned} \gamma (V)\theta _2^2(x)-c\theta _2(x)+a \le&\,\,\theta _2^2(x)-c\theta _2(x)+a \\ =&\,\,\theta _1(x)^2-c\theta _1(x)+a+\frac{1}{k_0}\left( \frac{1}{k_0}\lambda ^2+2\lambda \theta _1(x)-c\lambda \right) \nonumber \\ \le&\,\,\theta _1(x)^2-c\theta _1(x)+a+\frac{1}{2k_0}\lambda (2\lambda -c).\nonumber \end{aligned}$$

(5.3.12)

Moreover, owing to the fact $\lim _{x\rightarrow +\infty }\theta _1(x)=\lambda$, we have

$$\lim _{x\rightarrow +\infty }(\theta _1(x)^2-c\theta _1(x)+a)=\lambda ^2-c\lambda +a=0.$$

Then choosing $\delta$ sufficiently small, we obtain that for all $x>x_\delta$

$$\begin{aligned} \theta _1(x)^2-c\theta _1(x)+a\le \frac{1}{4k_0}\lambda (c-2\lambda ). \end{aligned}$$

(5.3.13)

Inserting (5.3.13) into (5.3.12), we obtain $\gamma (V)\theta _2^2(x)-c\theta _2(x)+a \le \frac{1}{4k_0}\lambda (2\lambda -c)<0.$ Thus, (5.3.10) follows and Lemma 5.7 is proved.

5.3.2 Auxiliary Problems

In this section, we shall investigate some auxiliary problems which act as bridges to our concerned problem.

1. An auxiliary parabolic problem

In the sequel, for convenience, we use $\gamma '(v)$ and $\gamma ''(v)$ to denote the first- and second-order derivatives of $\gamma (v)$ with respect to v, respectively. This should not be confused with $U', V', U'', V''$ where the prime $'$ means the differentiation with respect to x. Given $u\in \mathscr {E}_n$, we first consider the following equation:

$$\begin{aligned} V''+cV'+u-V=0 \end{aligned}$$

(5.3.14)

which, subject to variation of constants, yields

$$\begin{aligned} V:=V(x;u)=\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^xe^{\lambda _1(x-s)}u(s)ds+\int _x^{+\infty } e^{\lambda _2(x-s)}u(s)ds\right) . \end{aligned}$$

(5.3.15)

Now taking V in (5.3.15) as a known function, we define

$$\begin{aligned}&F(U,U') \\ :=&\,\,\frac{1}{\gamma (V)}\left\{ \left( 2\gamma '(V)V'+c\right) U'+\left[ \gamma ''(V)|V'|^2+\gamma '(V)(V-U-cV')+a\right] U-bU^2\right\} . \end{aligned}$$

By $U(x,t;u,\overline{U})$, we denote the solution of the following Cauchy problem:

$$\begin{aligned} \left\{ \begin{aligned}&U_t=U''+F(U,U'),&x \in \mathbb {R}, t>0 \\&U(x,0;u,\overline{U})=\overline{U}(x),&x \in \mathbb {R}. \end{aligned}\right. \end{aligned}$$

(5.3.16)

From Lemma 5.6 and the definition of $\gamma (\cdot )$, the boundedness of $\frac{1}{\gamma (V)}$, $\gamma '(V)$, $\gamma ''(V)$, V and $V'$ has been guaranteed. Then the comparison principle is applicable to (5.3.16). By the semigroup theory, U can be represented as

$$\begin{aligned} U(x,t;u,\overline{U})=&\,\,e^{t(\varDelta -1)}\overline{U}(x)+\int _0^te^{-(t-s)}e^{(t-s)\varDelta }(U+F(U,U'))(x,s)ds. \end{aligned}$$

The local existence of solutions to (5.3.16) can be obtained by the well-known fixed point theorem (cf. see Salako and Shen 2017b, Theorem 1.1) along with standard parabolic estimates. We omit the details here for brevity and assume that the solution of (5.3.16) exists in an maximal interval [0, T) for some $T\in (0, \infty ]$ with $U(x,0;u,\overline{U})>0$ for $x\in \mathbb R$. Then the comparison principle for (5.3.16) implies that $U(x,t;u,\overline{U})>0$ for all $(x,t)\in \mathbb {R}\times [0,T)$.

Proposition 5.1

If $c\ge 2\sqrt{a}$ and $b> b^*(m,a)$ with $b^*(m,a)$ defined in (5.1.12), there exists $\delta >0$ such that for any $u\in \mathscr {E}_n$, the solution $U(x,t;u,\overline{U})$ of (5.3.16) satisfies $U(\cdot ,t;u,\overline{U})\in \mathscr {E}_n$ for all $t\in [0,+\infty )$.

Proof

Denote

$$\begin{aligned}&L(U)\\ :=&\,\,\gamma (V)U''+\left( 2\gamma '(V)V'+c\right) U'+\left( \gamma ''(V)(V')^2+\gamma '(V)(V-U-cV')+a\right) U-bU^2\nonumber \end{aligned}$$

(5.3.17)

with V defined in (5.3.15). Noticing $\gamma (V)>0$, we have

$$U''+F(U,U')=\frac{L(U)}{\gamma (V)}.$$

Hence, a function U(x) is a super-solution (resp. sub-solution) of (5.3.16) if $L(U)\le 0$ (reps. $L(U)\ge 0$). Firstly, we need to prove that for any solution $u\in \mathscr {E}_n$, there exists $U(x,t;u,\overline{U})\le \overline{U}$. For any $s\ge 0$, from the definition of $\gamma (\cdot )$, we have

$$\begin{aligned} 0<\gamma (s)=\frac{1}{(1+s)^m}\le 1,\quad -m<\gamma '(s)=-\frac{m}{(1+s)^{m+1}}<0, \end{aligned}$$

(5.3.18)

and

$$\begin{aligned} 0<\gamma ''(s)=\frac{m(m+1)}{(1+s)^{m+2}}\le m(m+1). \end{aligned}$$

(5.3.19)

From (5.3.17), using (5.3.2), (5.3.3), (5.3.18) and (5.3.19), by the definition of $\eta$ in (5.2.9), it is easy to verify that

$$\begin{aligned} L(\eta )&=\left( \gamma ''(V)(V')^2+\gamma '(V)(V-\eta -cV')+a\right) \eta -b\eta ^2 \\&\le \left( \frac{m(m+1)}{1+a}\eta ^2+m\left( 1+\frac{2c}{\sqrt{c^2+4}}\right) \eta +a-b\eta \right) \eta \\&\le \left( \frac{m(m+1)}{1+a}\eta ^2+3m\eta +a-b\eta \right) \eta =0. \end{aligned}$$

On the other hand, from (5.3.17), using (5.3.2), (5.3.3), (5.3.18) and (5.3.19), we obtain

$$\begin{aligned}&L(e^{-\lambda x}) \\ =&\,\,\gamma (V)\lambda ^2e^{-\lambda x}-\left( 2\gamma '(V)V'+c\right) \lambda e^{-\lambda x} \\&+\left( \gamma ''(V)(V')^2e^{-\lambda x}+\gamma '(V)(V-e^{-\lambda x}-cV')\right) e^{-\lambda x}+ae^{-\lambda x}-be^{-2\lambda x} \\ \le&\,\,\lambda ^2e^{-\lambda x}+\frac{2m\lambda }{\sqrt{1+a}}e^{-2\lambda x}-c\lambda e^{-\lambda x}+\frac{m(m+1)}{1+a}e^{-4\lambda x}+\left( m+\frac{2cm}{\sqrt{4+c^2}}\right) e^{-2\lambda x} \\&+ae^{-\lambda x}-be^{-2\lambda x} \\ \le&\,\,(\lambda ^2-c\lambda +a)e^{-\lambda x}+\left( \frac{2m\sqrt{a}}{\sqrt{1+a}}+\frac{m(m+1) }{1+a}e^{-2\lambda x}+3m-b\right) e^{-2\lambda x}, \end{aligned}$$

where we have used the fact that $\lambda \in (0,\sqrt{a}]$. Noticing

$$b> b^*(m,a)>\frac{2m\sqrt{a}}{\sqrt{1+a}}+3m,$$

by choosing $\delta$ sufficiently small in (5.3.26), we obtain $L(e^{-\lambda x})\le 0$ for all $x>x_\delta$. By the comparison principle for parabolic equations, it follows that $U(x,t;u,\overline{U})\le \overline{U}$.

Now we prove that for any $u\in \mathscr {E}_n$, we have $U(x,t;u,\overline{U})\ge \underline{U}_n$. From (5.3.17), using (5.3.2), (5.3.3), (5.3.18) and (5.3.19), we obtain

$$\begin{aligned} L(\delta )&=\gamma ''(V)(V')^2\delta +\gamma '(V)(V-\delta -cV')\delta +\delta (a-b\delta )\nonumber \\&\ge \gamma '(V)(V-cV')\delta +\delta (a-b\delta ) \\&\ge \delta \left( a-b\delta -m\eta \left( 1+\frac{2c}{\sqrt{c^2+4}}\right) \right) \nonumber \\&\ge \delta \left( a-b\delta -3m\eta \right) .\nonumber \end{aligned}$$

(5.3.20)

Owing to the fact $b> b^*(m,a)$, we obtain

$$\begin{aligned} \eta =\frac{2a}{b-3m+\sqrt{\left( b-3m\right) ^2-\frac{4m(m+1)a}{1+a}}}<\frac{a}{3m}. \end{aligned}$$

(5.3.21)

Substituting (5.3.21) into (5.3.20), we have $L(\delta )\ge 0$ for sufficiently small $\delta$. On the other hand, using (5.3.17), by direct but tedious calculations, we have

$$\begin{aligned}&\quad L(d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x)x})\nonumber \\&=\gamma (V)(d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x)x})'' +(2\gamma '(V)V'+c)(d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x)x})'\nonumber \\&\quad +\left( \gamma ''(V)(V')^2+\gamma '(V)(V-(d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x)x})-cV')+a\right) \nonumber \\&\quad \cdot (d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x) x})-b(d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x)x}))^2 \\&\ge \left( \gamma (V)\theta _1^2(x)-c\theta _1(x)+a\right) d_ne^{-\theta _1(x)x} +\left( \gamma (V)\theta _2^2(x)-c\theta _2(x)+a\right) d_0e^{-\theta _2(x)x}\nonumber \\&\quad +d_ne^{-\theta _1(x)x}\left[ \gamma (V)\left( (\theta _1'(x)x)^2+2\theta _1'(x)\theta _1(x)x-\theta _1''(x)x-2\theta _1'(x)\right) -c\theta _1'(x)x\right. \nonumber \\&\quad \left. -2\gamma '(V)V'(\theta _1'(x)x+\theta _1(x))+\gamma ''(V)(V')^2+\gamma '(V)(V-cV')\right] \nonumber \\&\quad +d_0e^{-\theta _2(x)x}\left[ \gamma (V)\left( (\theta _2'(x)x)^2+2\theta _2'(x)\theta _2(x)x-\theta _2''(x)x-2\theta _2'(x)\right) -c\theta _2'(x)x\right. \nonumber \\&\quad \left. -2\gamma '(V)V'(\theta _2'(x)x+\theta _2(x))+\gamma ''(V)(V')^2+\gamma '(V)(V-cV')\right] \nonumber \\&\quad -b(d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x)x})^2.\nonumber \end{aligned}$$

(5.3.22)

To prove that $L(d_ne^{-\theta _1(x)x}+d_0e^{-\theta _2(x)x})\ge 0$, we consider the cases $c=2\sqrt{a}$ and $c>2\sqrt{a}$ separately.

Case 1. $c=2\sqrt{a}$. In this case, we have $d_0=1$ and substitute it into (5.3.22). Using (5.3.18) and Lemmas 5.1 and 5.7, by choosing sufficiently small $\delta$, for $x>x_\delta$, we obtain

$$\begin{aligned} \gamma '(V)<0,\quad \gamma ''(V)>0,\quad \theta _1'(x)>0,\quad \theta _1''(x)<0 \end{aligned}$$

and

$$\begin{aligned} \gamma (V)\theta _1^2(x)-c\theta _1(x)+a\ge 0,\quad \gamma (V)\theta _2^2(x)-c\theta _2(x)+a\ge \frac{a}{64}, \end{aligned}$$

from which we obtain that for any $x>x_\delta$,

$$\begin{aligned}&\quad L(d_ne^{-\theta _1(x)x}+e^{-\theta _2(x)x}) \\&\ge \frac{a}{64}e^{-\theta _2(x)x}+d_ne^{-\theta _1(x)x}\nonumber \\&\quad \cdot \left[ -2\gamma (V)\theta _1'(x)-c\theta _1'(x)x-2\gamma '(V)V'(\theta _1'(x)x+\theta _1(x))+\gamma '(V)(V-cV')\right] \nonumber \\&\quad +e^{-\theta _2(x)x}\left[ -2\gamma (V)\theta _2'(x)-c\theta _2'(x)x-2\gamma '(V)V'(\theta _2'(x)x+\theta _2(x))+\gamma '(V)(V-cV')\right] \nonumber \\&\quad -b(d_ne^{-\theta _1(x)x}+e^{-\theta _2(x)x})^2.\nonumber \end{aligned}$$

(5.3.23)

Furthermore, from (5.2.3) and Lemmas 5.1 and 5.6, we have

$$\begin{aligned} 0<\theta _1(x)<\sqrt{a},\quad 0<\theta _1'(x)\le 2K_1e^{-\frac{\lambda }{2} x} \end{aligned}$$

and

$$\begin{aligned} 0<V(x;u)\le \min \left\{ \frac{e^{-\lambda x}}{1+a},\eta \right\} ,\quad |V'(x;u)|\le \min \left\{ \frac{\eta }{\sqrt{1+a}},\frac{e^{-\lambda x}}{\sqrt{1+a}}\right\} . \end{aligned}$$

By the above estimates and (5.3.18), we arrive at the following estimates:

$$\begin{aligned}&-2\gamma (V)\theta _1'(x)-c\theta _1'(x)x-2\gamma '(V)V'(\theta _1'(x)x+\theta _1(x))+\gamma '(V)(V-cV') \\ \ge&\,\,-\left( 4+2cx+\frac{4m\eta x}{\sqrt{1+a}}\right) K_1e^{-\frac{\lambda }{2}x}-\left( \frac{2m\sqrt{a}}{\sqrt{1+a}}+\frac{m}{1+a} +\frac{cm}{\sqrt{1+a}}\right) e^{-\lambda x}\nonumber \\ =&\,\,-\left( 4+4\sqrt{a}x+\frac{4m\eta x}{\sqrt{1+a}}\right) K_1e^{-\frac{\lambda }{2} x}-\left( \frac{4m\sqrt{a}}{\sqrt{1+a}}+\frac{m}{1+a}\right) e^{-\lambda x}. \nonumber \end{aligned}$$

(5.3.24)

Then from the fact that $\theta _2(x)=\theta _1(x)+\frac{\lambda }{4}$, we get

$$\begin{aligned}&-2\gamma (V)\theta _2'(x)-c\theta _2'(x)x-2\gamma '(V)V'(\theta _2'(x)x+\theta _2(x))+\gamma '(V)(V-cV') \\ =&\,\,-2\gamma (V)\theta _1'(x)-c\theta _1'(x)x-2\gamma '(V)V'(\theta _1'(x)x+\theta _1(x))+\gamma '(V)(V-cV')\nonumber \\&-\frac{1}{2}\gamma '(V)V'\lambda \nonumber \\ \ge&\,\,-\left( 4+4\sqrt{a}x+\frac{4m\eta x}{\sqrt{1+a}}\right) K_1e^{-\frac{\lambda }{2} x}-\left( \frac{4m\sqrt{a}}{\sqrt{1+a}}+\frac{m}{1+a}+\frac{m\sqrt{a}}{2\sqrt{1+a}}\right) e^{-\lambda x}.\nonumber \end{aligned}$$

(5.3.25)

Substituting (5.3.24) and (5.3.25) into (5.3.23), we end up with

$$\begin{aligned}&L(d_ne^{-\theta _1(x)x}+e^{-\theta _2(x)x})\nonumber \\&\ge e^{-\theta _2(x)x}\bigg \{\frac{a}{64}-K_1\left( 4+4\sqrt{a}x+\frac{4m\eta x}{\sqrt{1+a}}\right) \left( e^{-\frac{\lambda }{2} x}+d_ne^{(\theta _2(x)-\theta _1(x)-\frac{\lambda }{2})x}\right) \\&-\left( \frac{4m\sqrt{a}}{\sqrt{1+a}}+\frac{m}{1+a}\right) \left( e^{-\lambda x}+d_ne^{(\theta _2(x)-\theta _1(x)-\lambda )x}\right) -\frac{m\sqrt{a}}{2\sqrt{1+a}}e^{-\lambda x}\nonumber \\&-b\left( d_n^2e^{(\theta _2(x)-2\theta _1(x))x}+e^{-\theta _2(x)x}+2d_ne^{-\theta _1(x)x}\right) \bigg \}.\nonumber \end{aligned}$$

(5.3.26)

From (5.2.5) and (5.2.6), we have $\theta _2(x)-2\theta _1(x)<0$ and $\theta _2(x)-\theta _1(x)-\lambda<\theta _2(x)-\theta _1(x)-\frac{\lambda }{2}<0$ for $x>x_\delta$, then for $c=2\sqrt{a}$, by choosing $\delta$ sufficiently small in (5.3.26), we obtain

$$L(d_ne^{-\theta _1(x)x}+e^{-\theta _2(x) x})\ge 0$$

for all $x>x_\delta$.

Case 2. $c>2\sqrt{a}$. Inserting $d_0=-1$ in (5.3.22), using Lemmas 5.1, 5.6 and 5.7, we obtain

$$\begin{aligned}&0<\theta _1(x)<\sqrt{a},\quad 0<\theta _1'(x)\le 2K_2e^{-\lambda x}\qquad 0>\theta _1''(x)\ge -2\lambda K_2e^{-\lambda x},\\&0<V(x;u)\le \min \left\{ \frac{e^{-\lambda x}}{1+a},\eta \right\} ,\\&|V'(x;u)| \le \min \left\{ \frac{2\eta }{\sqrt{c^2+4}},\frac{2e^{-\lambda x}}{\sqrt{c^2+4}}\right\} \le \min \left\{ \frac{\eta }{\sqrt{1+a}},\frac{e^{-\lambda x}}{\sqrt{1+a}}\right\} ,\\&\gamma (V)\theta _1^2(x)-c\theta _1(x)+a\ge 0,\quad \gamma (V)\theta _2^2(x)-c\theta _2(x)+a\le -\frac{\lambda (c-2\lambda )}{4k_0}. \end{aligned}$$

By these results, (5.3.18) and (5.3.19), for any $x>x_\delta$, noticing that $\theta _2(x)=\theta _1(x)+\frac{\lambda }{k_0}$, we obtain

$$\begin{aligned}&\quad L(d_ne^{-\theta _1(x)x}-e^{-\theta _2(x)x})\nonumber \\&\ge \left( \gamma (V)\theta _1^2(x)-c\theta _1(x)+a\right) d_ne^{-\theta _1(x)x} -\left( \gamma (V)\theta _2^2(x)-c\theta _2(x)+a\right) e^{-\theta _2(x)x}\nonumber \\&\quad +d_ne^{-\theta _1(x)x}\left[ -2\gamma (V)\theta _1'(x)-c\theta _1'(x)x-2\gamma '(V)V'(\theta _1'(x)x+\theta _1(x))+\gamma '(V)(V-cV')\right] \nonumber \\&\quad -e^{-\theta _2(x)x}\left[ \gamma (V)((\theta _2'(x)x)^2+2\theta _2'(x)\theta _2(x)x-\theta _2''(x)x)-2\gamma '(V)V'(\theta _2'(x)x+\theta _2(x))\right. \nonumber \\&\quad \left. +\gamma ''(V)(V')^2-c\gamma '(V)V'\right] -b(d_ne^{-\theta _1(x)x}-e^{-\theta _2(x)x})^2 \\&\ge e^{-\theta _2(x)x}\left\{ \frac{\lambda (c-2\lambda )}{4k_0}-b\left( d_n^2e^{(\theta _2(x)-2\theta _1(x))x}+e^{-\theta _2(x)x}\right) \right. \nonumber \\&\quad -\left( 4K_2+2cxK_2+\frac{2m}{\sqrt{1+a}}(2K_2e^{-\lambda x}x+\sqrt{a})+m\left( \frac{1}{1+a}+\frac{2c}{\sqrt{4+c^2}}\right) \right) \nonumber \\ {}&\quad \cdot d_ne^{(\theta _2(x)-\theta _1(x)-\lambda )x}\nonumber \\&\quad \left. -\left( (2K_2x)^2e^{-\lambda x}+4K_2\left( \sqrt{a}+\frac{\lambda }{k_0}\right) x+2\lambda K_2x+\frac{2m}{\sqrt{1+a}}\left( 2K_2xe^{-\lambda x}+\sqrt{a}+\frac{\lambda }{k_0}\right) \right. \right. \nonumber \\&\qquad \left. \left. +\frac{m(m+1)}{1+a}e^{-\lambda x}+\frac{2cm}{\sqrt{4+c^2}}\right) e^{-\lambda x}\right\} .\nonumber \end{aligned}$$

(5.3.27)

Noticing $\theta _2(x)-2\theta _1(x)<0$ and $\theta _2(x)-\theta _1(x)-\lambda <0$ for $x>x_\delta$, then for $c>2\sqrt{a}$, by choosing $\delta$ sufficiently small in (5.3.27), we obtain

$$L(d_ne^{-\theta _1(x)x}-e^{-\theta _2(x) x})\ge 0$$

for all $x>x_\delta$. Then by the comparison principle for parabolic equations, we obtain $U(x,t;u)\ge \underline{U}_n$ for $c\ge 2\sqrt{a}$.

Summing up, by choosing

$$\begin{aligned} \eta :=\frac{2a}{b-3m+\sqrt{\left( b-3m\right) ^2-\frac{4m(m+1)a}{1+a}}} \end{aligned}$$

(5.3.28)

and sufficiently small $\delta$, $(\overline{U},\underline{U}_n)$ is a pair of super- and sub-solutions of (5.3.16) (see a schematic of super- and sub-solutions illustrated in Fig. 5.2). Denoting $U(x,t;u,\overline{U})$ the unique solution of (5.3.16), by the comparison principle for parabolic equations, we obtain $\underline{U}_n\le U(x,t;u,\overline{U})\le \overline{U}$ and thus $U(x,t;u,\overline{U})\in \mathscr {E}_n$. This completes the proof of Proposition 5.1.

2. An auxiliary elliptic problem

Now for $u\in \mathscr {X}_0:=\bigcap _{n>1}\mathscr {E}_n$, we study the following problem:

$$\begin{aligned} \left\{ \begin{aligned}&\gamma (V)U''+\left( 2\gamma '(V)V'+c\right) U'+\left( \gamma ''(V)(V')^2+\gamma '(V)(V-U-cV')+a\right) U\\&-bU^2=0, \\&V''+cV'+u-V=0, \end{aligned} \right. \end{aligned}$$

(5.3.29)

which is equivalent to solving $L(U)=0$.

Proposition 5.2

For every $u\in \mathscr {X}_0$, if $c\ge 2\sqrt{a}$ and $b> b^*(m,a)$ with $b^*(m,a)$ defined in (5.1.12), denote $U(x,t;u,\overline{U})$ the solution of (5.3.16) with $U(x,0;u,\overline{U})=\overline{U}$, there exists a unique function $U(x;u)\in \mathscr {X}_0$ such that

$$U(x;u)=\lim _{t\rightarrow \infty }U(x,t;u,\overline{U})=\inf _{t>0}U(x,t;u,\overline{U})$$

and U(x; u) is the unique solution of (5.3.29) satisfying

$$\begin{aligned} \liminf _{x\rightarrow -\infty }U(x;u)>0\quad \hbox {and}\;\lim _{x\rightarrow +\infty }\frac{U(x;u)}{e^{-\lambda x}}=1. \end{aligned}$$

(5.3.30)

Proof

From Proposition 5.1, we have

$$\begin{aligned} U(x,t;u,\overline{U})\le \overline{U}(x)\quad \hbox {for all}\;(x,t)\in \mathbb R\times [0,+\infty ). \end{aligned}$$

(5.3.31)

For any $0\le t_1\le t_2$, noticing

$$U(x,t_2;u,\overline{U})=U(x,t_1;u,U(x,t_2-t_1;u,\overline{U})),$$

from (5.3.31), we have

$$U(x,t_2-t_1;u,\overline{U})\le \overline{U}(x).$$

Then using again the comparison principle for parabolic equations, we obtain

$$U(x,t_2;u,\overline{U})\le U(x,t_1;u,\overline{U}),$$

which implies that $U(x,\cdot ;u,\overline{U})$ is decreasing with respect to t. Noticing

$$U(x,\cdot ;u,\overline{U})$$

has lower and upper bounds since $U(x,\cdot ;u,\overline{U})\in \mathscr {E}_n$ as shown in Lemma 5.1, one can conclude that there exists a unique U(x; u) such that

$$\begin{aligned} U(x;u)=\lim _{t\rightarrow \infty }U(x,t;u,\overline{U})=\inf _{t>0}U(x,t;u,\overline{U}) \end{aligned}$$

(5.3.32)

for all $x\in \mathbb R$. Denote

$$U_n(x,t)=U(x,t+t_n;u,\overline{U})$$

for $(x,t)\in \mathbb R\times [0,\infty )$, where $\{t_n\}_{n\ge 1}$ is an increasing sequence of positive real numbers converging to $+\infty$. Then from the elliptic regularity theory for (5.3.14) and parabolic regularity theory for (5.3.16), we obtain that for all $1<p<\infty$, $R>0$, $T>0$,

$$\Vert V\Vert _{W^{2,p}(-R,R)}\le C\ \ \mathrm {and} \ \ \Vert U_n\Vert _{W^{2,1}_{p}((-R,R)\times (0,T))}\le C.$$

From the Sobolev embedding theorem, we obtain

$$\Vert V\Vert _{C_{\mathrm {loc}}^{1,\alpha }(\mathbb {R})}\le C \ \ \mathrm {and} \ \ \Vert U_n\Vert _{C^{\alpha ,\alpha /2}_{\mathrm {loc}}(\mathbb R\times (0,+\infty ))}\le C.$$

Arzelà–Ascoli’s theorem and Schauder’s theory for parabolic equation (cf. Krylov 1996) imply that there is a subsequence $\{U_{n'}\}_{n'\ge 1}$ of the sequence $\{U_{n}\}_{n\ge 1}$ and a function $\tilde{U}\in C^{2,1}(\mathbb R\times (0,\infty ))$, such that $\{U_{n'}\}_{n'\ge 1}$ converges to $\tilde{U}$ locally uniformly in $C^{2,1}(\mathbb R\times (0,\infty ))$ as $n'\rightarrow \infty$. Hence, $\tilde{U}(x,t)$ solves (5.3.29) and $\tilde{U}\in \mathscr {X}_0$. On the other hand, noticing $\tilde{U}(x,t)=\lim _{t\rightarrow \infty }U(x,t;u,\overline{U})$, from (5.3.32), we have $U(x;u)=\tilde{U}(x,t)$ for every $x\in \mathbb R$ and $t\ge 0$, from which we obtain that $U(x;u)\in \mathscr {X}_0$ is a solution of (5.3.29). Furthermore, from (5.2.10) and the definition of $\mathscr {X}_0$, we obtain

$$\begin{aligned} \liminf _{x\rightarrow -\infty }U(x;u)>0 \end{aligned}$$

and

$$\begin{aligned} d_n\le \liminf _{x\rightarrow +\infty }\frac{U(x;u)}{e^{-\lambda x}}\le \limsup _{x\rightarrow +\infty }\frac{U(x;u)}{e^{-\lambda x}}=1 \end{aligned}$$

(5.3.33)

for any $n\ge 2$. Noticing $\lim _{n\rightarrow \infty }d_n=1$, by taking $n\rightarrow \infty$ in (5.3.33), we obtain

$$\begin{aligned} \lim _{x\rightarrow +\infty }\frac{U(x;u)}{e^{-\lambda x}}=1. \end{aligned}$$

The uniqueness of U(x; u) satisfying (5.3.30) follows from the same arguments as that in Lemma 3.6 in Salako and Shen (2017a). The proof is thus completed.

5.3.3 Minimal Wave Speed

In this section, we shall prove Theorems 5.1 and 5.2. To this end, we first prove the following result concerning the asymptotic behavior of solutions to (5.1.10) as $z \rightarrow \pm \infty$.

Proposition 5.3

Assume that $a>0$ and $m>0$ satisfy (5.1.14). Then any solution $(U,V)\in (C^2(\mathbb R)\cap \mathscr {X}_0)^2$ to (5.1.10) has the property that

$$\lim _{z\rightarrow +\infty }U(z)=\lim _{z\rightarrow +\infty }V(z)=0,\quad \lim _{z\rightarrow -\infty }U(z) =\lim _{z\rightarrow -\infty }V(z)=a/b$$

and

$$\lim _{z\rightarrow \pm \infty }U'(z)=\lim _{z\rightarrow \pm \infty }V'(z)=0.$$

Proof

From the fact that $(U,V)\in \mathscr {X}_0^2$ and Lemma 5.6, we obtain

$$\begin{aligned} |U(z)|\le \eta , \quad |V(z)|\le \eta \quad \hbox {and}\; |V'(z)|\le \frac{\eta }{\sqrt{1+a}} \end{aligned}$$

(5.3.34)

for all $z\in \mathbb R$. From the first equation of (5.1.10), by the Hölder regularity estimates for bounded solutions of elliptic equations and the Schauder theory (Gilbarg and Trudinger 2001), there exists $C>0$ independent of z and $\alpha \in (0,1)$ such that $\Vert U\Vert _{C^{2,\alpha }(z,z+1)}\le C$ and $\Vert V\Vert _{C^{2,\alpha }(z,z+1)}\le C$ for all $z\in \mathbb R$, from which it follows that

$$\begin{aligned} |U'(z)|\le C, \quad |U''(z)|\le C\quad \hbox {and}\;|V''(z)|\le C \end{aligned}$$

(5.3.35)

for all $z\in \mathbb R$. Multiplying the first equation of (5.1.10) by $(a-bU)$, integrating over $[-R,R]$, we obtain

$$\begin{aligned} 0=&\,\,\int _{-R}^R(\gamma (V)U)''(a-bU)dz+c\int _{-R}^RU'(a-bU)dz+\int _{-R}^R U(a-bU)^2dz \\ =&\,\,(\gamma (V)U)'(a-bU)\big |_{z=-R}^{z=R}+b\int _{-R}^R(\gamma '(V)V'U+\gamma (V)U')U'dz +caU\big |_{z=-R}^{z=R} \\&-\frac{1}{2}cbU^2\big |_{z=-R}^{z=R}+\int _{-R}^R U(a-bU)^2dz. \end{aligned}$$

Then using (5.3.18), (5.3.34) and (5.3.35), we find a constant $C_1$ independent of R such that

$$\begin{aligned}&\frac{b}{(1+\eta )^m}\int _{-R}^R|U'|^2dz+\int _{-R}^R U(a-bU)^2dz\nonumber \\ \le&\,\,b\int _{-R}^R\gamma (V)|U'|^2dz+\int _{-R}^R U(a-bU)^2dz \\ \le&b\int _{-R}^R|\gamma '(V)V'UU'|dz-(\gamma (V)U)'(a-bU)\big |_{z=-R}^{z=R}-caU\big |_{z=-R}^{z=R}+\frac{1}{2}cbU^2\big |_{z=-R}^{z=R}\nonumber \\ \le&\,\,C_1+\frac{1}{2}bm\eta \left( \int _{-R}^R|U'|^2dz+\int _{-R}^R|V'|^2dz\right) .\nonumber \end{aligned}$$

(5.3.36)

On the other hand, multiplying the second equation of (5.1.10) by $V''$ and integrating the result over $[-R,R]$, we obtain

$$\begin{aligned} 0&=\int _{-R}^R|V''|^2dz+c\int _{-R}^RV'V''dz+\int _{-R}^RUV''dz-\int _{-R}^RVV''dz \\&=\int _{-R}^R|V''|^2dz+\frac{c}{2}(V')^2\big |_{z=-R}^{z=R}+UV'\big |_{z=-R}^{z=R} -\int _{-R}^RU'V'dz-VV'\big |_{z=-R}^{z=R}+\int _{-R}^R|V'|^2dz. \end{aligned}$$

This along with (5.3.34) and (5.3.35) yields

$$\begin{aligned} \int _{-R}^R|V''|^2dz+\int _{-R}^R|V'|^2dz&\le C_2+\int _{-R}^RU'V'dz \le C_2+\frac{1}{2}\int _{-R}^R|U'|^2dz+\frac{1}{2}\int _{-R}^R|V'|^2dz, \end{aligned}$$

where $C_2$ is a constant independent of R. Then it follows that

$$\begin{aligned} \int _{-R}^R|V'|^2dz\le 2C_2+\int _{-R}^R|U'|^2dz. \end{aligned}$$

(5.3.37)

Substituting (5.3.37) into (5.3.36), one can find a constant $C_3=C_1+bm\eta C_2$ independent of R such that

$$\begin{aligned} \frac{b}{(1+\eta )^m}\int _{-R}^R|U'|^2dz+\int _{-R}^R U(a-bU)^2dz \le C_3+bm\eta \int _{-R}^R|U'|^2dz. \end{aligned}$$

(5.3.38)

Note that (5.3.28) together with condition (5.1.14) implies

$$\begin{aligned} \frac{1}{(1+\eta )^m}-m\eta >0. \end{aligned}$$

Sending $R\rightarrow \infty$ in (5.3.38), we obtain

$$\begin{aligned} b\left( \frac{1}{(1+\eta )^m}-m\eta \right) \int _{\mathbb R}|U'|^2dz+\int _{\mathbb R} U(a-bU)^2dz \le C_3. \end{aligned}$$

(5.3.39)

By sending $R\rightarrow \infty$ in (5.3.37), we find a constant $C_4>0$ such that

$$\begin{aligned} \int _{\mathbb R}|V'|^2dz\le C_4. \end{aligned}$$

(5.3.40)

Then (5.3.39) and (5.3.40) assert that

$$\begin{aligned} U'\in L^2(\mathbb R),\ \ U(a-bU)^2\in L^1(\mathbb R), \ \ V'\in L^2(\mathbb R). \end{aligned}$$

(5.3.41)

From (5.3.35) and (5.3.41), we obtain

$$\begin{aligned} \lim _{z\rightarrow \pm \infty }U(z)\in \{0,a/b\},\quad \lim _{z\rightarrow \pm \infty }U'(z)=0\quad \hbox {and}\; \lim _{z\rightarrow \pm \infty }V'(z)=0. \end{aligned}$$

(5.3.42)

Furthermore, from the definition of $\mathscr {X}_0$ and the fact that $U\in \mathscr {X}_0$, we obtain

$$\lim _{z\rightarrow +\infty }U(z)=0\quad \hbox {and}\; \lim _{z\rightarrow -\infty }U(z)=a/b.$$

On the other hand, from the second equation of (5.1.10), we have

$$\begin{aligned} V(z)=\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^ze^{\lambda _1(z-s)}U(s)ds+\int _z^{+\infty } e^{\lambda _2(z-s)}U(s)ds\right) \end{aligned}$$

(5.3.43)

with $\lambda _1<0$ and $\lambda _2>0$ defined in (5.3.4). Applying L’Hopital’s rule to (5.3.43), from the fact (5.3.42), we obtain

$$\begin{aligned} \lim _{z\rightarrow +\infty }V(z)&=\lim _{z\rightarrow +\infty }\frac{1}{\lambda _2-\lambda _1} \left( \frac{\int _{-\infty }^ze^{-\lambda _1s}U(s)ds}{e^{-\lambda _1z}} +\frac{\int _z^{+\infty }e^{-\lambda _2s}U(s)ds}{e^{-\lambda _2z}}\right) \\&=\frac{1}{\lambda _2-\lambda _1}\lim _{z\rightarrow +\infty }\left( \frac{U(z)}{-\lambda _1}+\frac{U(z)}{\lambda _2}\right) \\&=\lim _{z\rightarrow +\infty }U(z)=0 \end{aligned}$$

and

$$\begin{aligned} \lim _{z\rightarrow -\infty }V(z)&=\lim _{z\rightarrow -\infty }\frac{1}{\lambda _2-\lambda _1}\left( \frac{\int _{-\infty }^ze^{-\lambda _1s}U(s)ds}{e^{-\lambda _1z}} +\frac{\int _z^{+\infty }e^{-\lambda _2s}U(s)ds}{e^{-\lambda _2z}}\right) \\&=\frac{1}{\lambda _2-\lambda _1}\lim _{z\rightarrow -\infty }\left( \frac{U(z)}{-\lambda _1}+\frac{U(z)}{\lambda _2}\right) \\&=\lim _{z\rightarrow -\infty }U(z)=\frac{a}{b}. \end{aligned}$$

This completes the proof.

1. Proof of Theorem 5.1.  Note that a fixed point of the mapping $u\ni \mathscr {X}_0\mapsto U(\cdot ,u)\in \mathscr {X}_0$ formed in (5.3.29) is a solution to the wave equations (5.1.10). Hence, to prove the existence of traveling wave solutions to (5.1.6), it suffices to prove that the mapping $u\ni \mathscr {X}_0\mapsto U(\cdot ,u)\in \mathscr {X}_0$ formed in (5.3.29) has a fixed point. We shall achieve this by the Schauder fixed point theorem.

First, we prove that the mapping $u\ni \mathscr {X}_0\mapsto U(\cdot ,u)\in \mathscr {X}_0$ is compact. Let $\{u_n\}_{n\ge 1}$ be a sequence in $\mathscr {X}_0$. Denote $U_n=U(\cdot ,u_n)$, we have $U_n\in \mathscr {X}_0$. From the elliptic regularity theorem, we have that $\Vert U_n\Vert _{W^{2,p}_{\mathrm {loc}}(\mathbb R)}\le C$ for all $p>1$. From the Sobolev embedding theorem, we obtain $\Vert U_n\Vert _{C^{\alpha }_{\mathrm {loc}}(\mathbb R)}\le C,$ which along with Arzela–Ascoli’s theorem implies that there is a subsequence $\{U_{n'}\}_{n'\ge 1}$ of the sequence $\{U_{n}\}_{n\ge 1}$ and a function $U(x)\in C(\mathbb R)$, such that $\{U_{n'}\}_{n'\ge 1} \rightarrow U(x)$ locally uniformly in $C(\mathbb R)$. Furthermore, we have $U(x)\in \mathscr {X}_0$. Then the mapping $u\ni \mathscr {X}_0\mapsto U(\cdot ,u)\in \mathscr {X}_0$ is compact.

Second, we prove that the mapping $u\ni \mathscr {X}_0\mapsto U(\cdot ;u)\in \mathscr {X}_0$ is continuous. To this end, denote

$$\Vert u\Vert _*=\sum _{n=1}^\infty \frac{1}{2^n}\Vert u\Vert _{L^\infty ([-n,n])}.$$

Then any sequence of functions in $\mathscr {X}_0$ is convergent with respect to norm $\Vert \cdot \Vert _*$ if and only if it converges locally uniformly on $\mathbb R$. Let $u\in \mathscr {X}_0$ and $\{u_n\}_{n\ge 1}$ be a sequence in $\mathscr {X}_0$ such that $u_n$ converges to u locally uniformly on $\mathbb R$ as $n\rightarrow \infty$. Then by the elliptic regularity theorem applied to the second equation of (5.3.29) and the Sobolev embedding theorem, we obtain

$$\Vert V(\cdot ;u_n)\Vert _{C^{1,\alpha }_{\mathrm {loc}}(\mathbb R)}\le C.$$

From Arzelà–Ascoli’s theorem, there exists a subsequence of $\{V(\cdot ;u_{n})\}_{n\ge 1}$, still denoted by itself without confusion, such that

$$\lim _{n'\rightarrow \infty }V(\cdot ;u_{n})=V(\cdot ;u)\quad \hbox {in}\; C^1_{\mathrm {loc}}(\mathbb R).$$

Suppose by contradiction that the mapping $u\ni \mathscr {X}_0\mapsto U(\cdot ;u)\in \mathscr {X}_0$ is not continuous, then there exist $\delta >0$ and a subsequence $\{u_{n'}\}_{n'\ge 1}$ such that

$$\begin{aligned} \Vert U(\cdot ; u_{n'})-U(\cdot ; u)\Vert _*\ge \delta ,\quad \forall n\ge 1. \end{aligned}$$

(5.3.44)

By Schauder’s theory (Krylov 1996) applied to the first equation of (5.3.29) and the Sobolev embedding theorem, from Arzelà–Ascoli’s theorem, there is a subsequence $\{U(\cdot ; u_{n''})\}_{n''\ge 1}$ of the sequence $\{U(\cdot ;u_{n'})\}_{n'\ge 1}$ and a function $U(\cdot )\in C^{2}(\mathbb R)$, such that $\{U(\cdot ; u_{n''})\}_{n''\ge 1}$ converges to $U(\cdot )$ in $C^2_{\mathrm {loc}}(\mathbb R)$ and U is a solution of (5.3.29). Moreover, from the fact that $U(\cdot ;u_{n''})\in \mathscr {X}_0$ and

$$\lim _{n\rightarrow \infty }\Vert U(\cdot ;u_{n''})- U(\cdot )\Vert _*=0,$$

we obtain $U(\cdot )\in \mathscr {X}_0$. Then from Proposition 5.2, we obtain $U(\cdot )=U(\cdot ,u).$ By (5.3.44), then

$$\Vert U(\cdot ;u)-U(\cdot )\Vert _*\ge \delta ,$$

which is a contradiction. Hence, the mapping $u\ni \mathscr {X}_0\mapsto U(\cdot ;u)\in \mathscr {X}_0$ is continuous.

Now by Schauder’s fixed point theorem, there is $U\in \mathscr {X}_0$ such that $U(\cdot )=U(\cdot ;U)$. Denote $V(\cdot ):=V(\cdot ;U)$. Then (U, V) is a solution of (5.1.10). From the definition of $\mathscr {X}_0$ and (5.2.10), we obtain

$$\lim _{z\rightarrow +\infty }\frac{U(z)}{e^{-\lambda z}}=1.$$

This along with (5.3.5)–(5.3.6) and L’Hôpital’s Rule yields

$$\begin{aligned} \lim _{z\rightarrow +\infty }\frac{V(z)}{e^{-\lambda z}}&=\lim _{z\rightarrow +\infty }\frac{1}{\lambda _2-\lambda _1}\left( \frac{\int _{-\infty }^ze^{-\lambda _1s}U(s)ds}{e^{-(\lambda _1+\lambda )z}} +\frac{\int _z^{+\infty }e^{-\lambda _2s}U(s)ds}{e^{-(\lambda _2+\lambda )z}}\right) \\&=\frac{1}{\lambda _2-\lambda _1}\lim _{z\rightarrow +\infty }\left( \frac{U(z)}{-(\lambda _1+\lambda )e^{-\lambda z}}-\frac{U(z)}{-(\lambda _2+\lambda )e^{-\lambda z}}\right) =\frac{1}{1+a}. \end{aligned}$$

Since $U\in \mathscr {X}_0$, it follows that $\liminf \limits _{z\rightarrow -\infty }U(z)>0$. On the other hand, noticing for $z<x_\delta$, $U(z)>\delta$ and then

$$\begin{aligned} V(z)&=\frac{1}{\lambda _2-\lambda _1}\left( \int _{-\infty }^ze^{\lambda _1(z-s)}U(s)ds +\int _z^{+\infty }e^{\lambda _2(z-s)}U(s)ds\right) \\&\ge \frac{\delta }{\lambda _2-\lambda _1}\int _{-\infty }^ze^{\lambda _1(z-s)}ds=\frac{\delta }{(\lambda _2-\lambda _1)(-\lambda _1)}>0, \end{aligned}$$

from which $\liminf \limits _{z\rightarrow -\infty }V(z)>0$ follows. Finally, by the assumption (5.1.14) and Proposition 5.3, we finish the proof of Theorem 5.1.

2. Proof of Theorem 5.2.  Arguing by contradiction, for $c<2\sqrt{a}$, we suppose that there is a traveling wave solution $(u(x,t),v(x,t))=(U(x\cdot \xi -ct),V(x\cdot \xi -ct))$ of (5.1.6) connecting the constant solutions (a/b, a/b) and (0, 0). Take a sequence $\{z_n\}$ with $z_n\rightarrow +\infty$, then

$$\lim _{n\rightarrow +\infty }U(z_n)=\lim _{n\rightarrow +\infty }V(z_n)=\lim _{n\rightarrow +\infty }V'(z_n)=0.$$

Now we set

$$h_n(z)=\frac{U(z+z_n)}{U(z_n)},\quad U_n(z)=U(z+z_n),\quad V_n(z)=V(z+z_n).$$

As U is bounded and satisfies (5.1.10), the Harnack inequality implies that the shifted functions $U_n(z)$, $V_n(z)$ and $V_n'(z)$ converge to zero locally uniformly in z and the sequence $h_n$ is locally uniformly bounded and satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\gamma ''(V_n)(V'_n)^2h_n+\gamma '(V_n)(V_n-U_n-cV_n')h_n+2\gamma '(V_n)V_n'h_n'+\gamma (V_n)h_n''\\&+ch_n'+h_n(a-bU_n)=0,\\&V_n''+U_n-V_n+cV_n'=0 \end{aligned} \right. \end{aligned}$$

in $\mathbb R$. Thus, up to a subsequence, the sequence $\{h_n\}_{n\ge 1}$ converges to a function h that satisfies

$$\begin{aligned} h''+ch'+ah=0\quad \hbox {in}\;\mathbb R. \end{aligned}$$

(5.3.45)

Moreover, h is nonnegative and $h(0)=1$. Equation (5.3.45) admits such a solution if and only if $c\ge 2\sqrt{a}$, which leads to a contradiction. This denies our assumption and hence (5.1.6) admits no traveling wave solution connecting (a/b, a/b) and (0, 0) with speed $c<2\sqrt{a}$.

5.3.4 Selection of Wave Profiles

By introducing some auxiliary problems and spatially inhomogeneous relaxed decay rates for super- and sub-solutions constructed, we manage to establish the existence of traveling wavefront solutions to the density-suppressed motility system (5.1.6) with decay motility function (5.1.7), where we find that there is a minimal wave speed coincident with the one for the cornerstone Fisher-KPP equation and a maximum wave speed c resulting from the nonlinear diffusion. However, we are unable to characterize further properties of wave profiles such as monotonicity, stability and so on. In this section, we shall discuss the selection of possible wave profiles motivated by some argument in Ou and Yuan (2009).

1. Trailing edge wave profiles

In the spatially homogeneous situation, the system (5.1.6) has equilibria (0, 0) and (a/b, a/b), which are unstable saddle and stable node, respectively. This suggests that we should look for traveling wavefront solutions to (5.1.6) connecting (a/b, a/b) to (0, 0) as we have done. Now we linearize the ODE system (5.1.10) at the origin (0, 0) and let $U'=X, V'=Y$. Then we get the following linear system of (U, X, V, Y):

$$\begin{aligned} \begin{pmatrix} U'\\ X'\\ V'\\ Y' \end{pmatrix} = \begin{pmatrix} 0 &{} 1 &{} 0 &{} 0\\ -\frac{a}{\gamma (0)} &{} -\frac{c}{\gamma (0)} &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 1\\ -1 &{} 0 &{} 1 &{} -c \end{pmatrix} \begin{pmatrix} U\\ X\\ V\\ Y \end{pmatrix}. \end{aligned}$$

The eigenvalue $\lambda$ of the above coefficient matrix is

$$\Big (\lambda ^2+\frac{c}{\gamma (0)}\lambda +\frac{a}{\gamma (0)}\Big ) \Big (\lambda ^2+c\lambda -1 \Big )=0.$$

To ensure there is a positive trajectory connecting the equilibria (0, 0) and (a/b, a/b), we need to rule out the case that (0, 0) is a spiral, which amounts to require

$$\begin{aligned} c\ge 2 \sqrt{\gamma (0) a}. \end{aligned}$$

(5.3.46)

With $\gamma (v)$ given in (5.1.7), $\gamma (0)=1$ and (5.3.46) is equivalent to $c\ge 2 \sqrt{a}$. This is well consistent with our results obtained in Theorems 5.1 and 5.2. Under the restriction (5.3.46), it can be easily checked that the origin (0, 0) is either a stable node or saddle point, which indicates that the traveling wave profile around the origin (0, 0) will not be oscillatory or periodic.

Next, we linearize the system (5.1.10) at (a/b, a/b) and arrive at the following linearized system:

$$\begin{aligned} \begin{pmatrix} U'\\ X'\\ V'\\ Y' \end{pmatrix} = \begin{pmatrix} 0 &{} 1 &{} 0 &{} 0\\ \frac{a(b+\sigma _2)}{\sigma _1 b} &{} -\frac{c}{\sigma _1} &{} -\frac{a \sigma _2}{b \sigma _1} &{} \frac{a \sigma _2 c}{b \sigma _1}\\ 0 &{} 0 &{} 0 &{} 1\\ -1 &{} 0 &{} 1 &{} -c \end{pmatrix} \begin{pmatrix} U\\ X\\ V\\ Y \end{pmatrix} \end{aligned}$$

(5.3.47)

where $\sigma _1=\gamma (a/b) and \sigma _2=\gamma '(a/b)$. By some tedious computation, we find that the eigenvalue $\lambda$ of the above coefficient matrix is determined by the following characteristic equation:

$$\begin{aligned} \lambda ^4+\Big (c+\frac{c}{\sigma _1}\Big )\lambda ^3+\bigg (\frac{c^2}{\sigma _1}-\frac{a(b+\sigma _2)}{\sigma _1 b}-1\bigg )\lambda ^2-\frac{(a+1)c}{\sigma _1}\lambda +\frac{a}{\sigma _1}=0. \end{aligned}$$

(5.3.48)

We suppose that there are periodic solutions near the positive equilibrium (a/b, a/b), namely the above characteristic equation has purely imaginary roots $\lambda = \pm \omega i$, where $\omega$ is a real number. Then the substitution of this ansatz into the equation (5.3.48) immediately yields a necessary condition $c=0$, and consequently, we get

$$\begin{aligned} \omega ^4-\bigg (\frac{a(b+\sigma _2)}{\sigma _1 b}+1\bigg )\omega ^2+\frac{a}{\sigma _1}=0. \end{aligned}$$

(5.3.49)

Notice that $\sigma _2=\gamma '(a/b)<0$. Then a necessary and sufficient condition warranting that Eq. (5.3.49) has a real root $\omega$ is

$$\begin{aligned} |\sigma _2|<\frac{b}{a}\sigma _1 \bigg (\sqrt{\frac{a}{\sigma _1}}-1\bigg )^2. \end{aligned}$$

(5.3.50)

That is, the linearized system (5.3.47) at the equilibrium (a/b, a/b) will have periodic solutions if the condition (5.3.50) is fulfilled. Thereof, we anticipate that the non-monotone traveling wave solutions oscillating about the critical point (a/b, a/b) may exist, but whether the condition (5.3.50) is sufficient to guarantee that the nonlinear system (5.1.6) has similar oscillatory behavior around the equilibrium (a/b, a/b) is very hard to determine and even to predict due to the complexity induced by the nonlinear diffusion and cross-diffusion in the system. Below we shall use numerical simulations to illustrate that indeed the condition (5.3.50) plays a critical role for the nonlinear system in determining the monotonicity of wave profiles.

We consider the motility function $\gamma (v)=\frac{1}{(1+v)^m} (m>0)$ as given in (5.3.29). With simple calculation, we find that the condition (5.3.50) amounts to

$$\begin{aligned} \sqrt{m}<\sqrt{\frac{1+\vartheta }{\vartheta }} \ \big | \sqrt{a(1+\vartheta )^m}-1\big |,\ \ \vartheta =\frac{a}{b}. \end{aligned}$$

(5.3.51)

Without loss of generality, we first choose $m=6$ and $a=b=0.1$. Then $\vartheta =1$ and

$$\sqrt{\frac{1+\vartheta }{\vartheta }} \ \Big | \sqrt{a(1+\vartheta )^m}-1\Big |=2.1635<\sqrt{6}=2.4495.$$

Hence, the condition (5.3.51) is violated and no oscillation around $(a/b,a/b)=(1,1)$ is expected for the linearized system. To verify if this is the case for the nonlinear system (5.1.6), we set the initial value $(u_0, v_0)$ as

$$\begin{aligned} u_0(x)=v_0(x)=\frac{1}{1+e^{2(x-20)}} \end{aligned}$$

(5.3.52)

and perform the numerical simulations in an interval [0, 200] with Neumann boundary conditions to comply with the experiment. The numerical solution of (5.1.6) is shown in Fig. 5.3 where we observe that the solution will stabilize into monotone traveling waves although it oscillates initially. This is also well consistent with our analytical results about the existence of traveling wave solutions given in Theorem 5.1 when $\mathscr {K}(m,a)=0.4143<1$ if $m=6$ and $a=b=0.1$. Next, we choose $m=4$ and $a=b=1$ such that $\sqrt{\frac{1+\vartheta }{\vartheta }} \ \Big | \sqrt{a(1+\vartheta )^m}-1\Big |=4.2426$ and hence (5.3.51) holds. But numerically we still find that the system (5.1.6) will generate monotone traveling waves qualitatively similar to the patterns shown in Fig. 5.3 (not shown here for brevity). This implies that the condition (5.3.50) is not sufficient to induce non-monotone traveling waves oscillating around (a/b, a/b).

Fig. 5.3

Numerical simulations of wave propagation generated by the system (5.1.6) in [0, 200] with $\gamma (v)=\frac{1}{(1+v)^m}$ with $m=6$, $a=b=0.1, u_0=v_0=\frac{1}{1+e^{2(x-20)}}$

Now an important question is whether the density-suppressed motility system (5.1.6) is capable of producing persistent oscillating traveling waves to interpret (at least qualitatively) the pattern observed in the experiment (see Fig. 5.1). To explore this question numerically, we consider the following sigmoid motility function:

$$\begin{aligned} \gamma (v)=1-\frac{v-1}{\sqrt{0.1+(v-1)^2}} \end{aligned}$$

which decays but changes the convexity at the point $v=1$, in contrast to the decreasing function (5.1.7) whose convexity remains unchanged. We perform the numerical simulations for (5.1.6) with $a=b=0.2$ in an interval [0, 200] with the same initial value (5.3.52). Remarkably, we find non-monotone traveling wavefronts develop (see Fig. 5.4) and persist in time, where the wave oscillates at the trailing edge and propagates into the far field as time evolves. This is a prominent feature different from the patterns shown in Fig. 5.3 generated from the motility function (5.1.7). If we choose some other forms of decreasing function $\gamma (v)$ that changes its convexity at $v=a/b=1$, we shall numerically find similar non-monotone traveling wavefront patterns generated by (5.1.6).

The above numerical simulations indicate, although not proved in Chapter 5, that the density-suppressed motility system (5.1.6) can generate both monotone and non-monotone traveling wavefront solutions connecting (a/b, a/b) to (0, 0). It numerically appears that the change of convexity of $\gamma (v)$ at $v=a/b$ is necessary to generate the non-monotone traveling wavefronts oscillating at the trailing edge around the equilibrium (a/b, a/b). The underlying mechanism remains mysterious and we will leave it as an open question for future study.

Fig. 5.4

Numerical simulations of wave propagation generated by the system (5.1.6) in [0, 200] with $\gamma (v)=1-\frac{v-1}{\sqrt{0.1+(v-1)^2}}, a=b=0.2, u_0=v_0=\frac{1}{1+e^{2(x-20)}}$

Fig. 5.5

Snapshot of numerical simulations of outward expanding ring patterns in a disk generated by the system (5.1.6) with $\gamma (v)=\frac{1}{(1+v)^6}, a=b=0.1, u_0=v_0=4+e^{-(x^2+y^2)}$

Next, we are devoted to exploring the patterns in a disk to mimic the apparatus used in the experiment of Liu (2011) where the experiment was conducted in Petri dishes with bacteria initially inoculated at the center (see Fig. 5.1). In the numerical simulations, we set the domain as a disk with radius 10 and initially place the initial value $(u_0, v_0)=(4+e^{-(x^2+y^2)},4+e^{-(x^2+y^2)})$ in the center. We use the motility function given in (5.1.7) with $m=6$ and set out Neumann boundary (i.e., zero-flux) conditions aligned with the experiment reality. The snapshots of numerical patterns are recorded in Fig. 5.5, where we do observe the outward expanding ring patterns qualitatively analogous to the experiment patterns shown in Fig. 5.1. This validates the capability of model (5.1.6) to reproduce the experimental patterns. However, we should underline that it appears that the generation of oscillating patterns in two dimensions does not rely on the change of convexity of the motility function $\gamma (v)$ as shown in Fig. 5.5, which is very different from the situation in 1D as shown in Figs. 5.3 and 5.4. This imposes another interesting question elucidating this subtle difference.

2. Leading edge wave speeds

Following the spirit of classical method as in Mollison (1977) and Murray (2001), we discuss the selection of the wave speed c from the initial conditions given at infinity. Suppose that the initial value $(u_0, v_0)$ of the system (5.1.6) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} u_0(x) \sim Ae^{-\lambda x}, \\ v_0(x) \sim Be^{-\lambda x}, \end{array}\right. } \ \mathrm {as} \ x\ \rightarrow \infty \end{aligned}$$

(5.3.53)

with positive amplitudes A and B. Now we look for traveling wave solutions of (5.1.10) at the leading edge (i.e., $x \rightarrow \infty$) in the form of

$$\begin{aligned} {\left\{ \begin{array}{ll} u(x,t) \sim Ae^{-\lambda (x-ct)}, \\ v(x,t) \sim Be^{-\lambda (x-ct)}. \end{array}\right. } \end{aligned}$$

(5.3.54)

We substitute (5.3.54) into the first equation of (5.1.6) and get the dispersion relation between the wave speed c and the initial decay rate $\lambda$:

$$\begin{aligned} c=\gamma (0)\lambda +\frac{a}{\lambda }. \end{aligned}$$

(5.3.55)

Hence, by the standard argument as in Murray (2001), the asymptotic wave speed c of traveling wave solutions to (5.1.6) satisfies

$$\begin{aligned} c={\left\{ \begin{array}{ll} \gamma (0)\lambda +\frac{a}{\lambda }, &{} \text {if} \ \ 0<\lambda <\sqrt{a}, \\ 2 \sqrt{\gamma (0)a}, &{} \text {if}\ \ \lambda \ge \sqrt{a}. \end{array}\right. } \end{aligned}$$

(5.3.56)

Next, we plug (5.3.54) into the second equation of (5.1.6) and get the following relation on the amplitude of u and v:

$$\begin{aligned} A=[1+a+(\gamma (0)-1)\lambda ^2]B. \end{aligned}$$

(5.3.57)

Therefore, given the initial condition (5.3.53), the leading edge of traveling waves is fully determined by the ansatz (5.3.54) with wave speed (5.3.56) and amplitudes fulfilling (5.3.57).

As an example, we consider the motility function (5.1.7) chosen in the first part of this chapter, where $\gamma (0)=1$ and hence (5.3.55) gives

$$\lambda ^2-c \lambda +a=0$$

which is exactly the same as the equation (5.2.2). Furthermore, (5.3.57) gives $A=(1+a)B$ which well agrees with the result (5.1.13) in Theorem 5.1.

5.4 Asymptotic Behavior of Solutions to a Signal-Suppressed Motility Model

5.4.1 Space–Time $L^1$-Estimates for $u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }$

In this section, taking advantage of the special structure of the diffusive processes in (5.2.20) (also (5.1.16)), the classical duality arguments (cf. Tao and Winkler 2017a; Cañizo et al. 2014) are used to obtain the fundamental regularity information for a bootstrap argument. To this end, we denote by A the self-adjoint realization of $-\varDelta +1$ under homogeneous Neumann boundary condition in $L^2(\varOmega )$ with its domain given by $D(A)=\left\{ \varphi \in W^{2,2}(\varOmega )|\frac{\partial \varphi }{\partial \nu }=0\right\}$ and A is self-adjoint and possesses a family $(A^{\beta })_{\beta \in \mathbb {R}}$ of corresponding densely defined self-adjoint fractional powers.

Lemma 5.8

Assume that $m> 1$ and $D\ge 1$, then for $t>0$

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^2+\int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\le C\int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1}+C \end{aligned}$$

(5.4.1)

with constant $C>0$ independent of D.

Proof

Due to $\partial _t(u_{\varepsilon }+1)=u_{\varepsilon t}$, the first equation in (5.2.20) can be written as

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt} A^{-1}(u_{\varepsilon }+1)+\varepsilon (u_{\varepsilon }+1)^{M} +u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1} v_\varepsilon ^{-\alpha }\\ =&\,\,A^{-1}\left\{ \varepsilon (u_{\varepsilon }+1)^M+u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}v_\varepsilon ^{-\alpha }+ \beta u_{\varepsilon } f(w_{\varepsilon })\right\} . \end{aligned} \end{aligned}$$

(5.4.2)

Testing (5.4.2) by $u_{\varepsilon }+1$, one has

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{2}\frac{d}{dt}\int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^2+\varepsilon \int _{\varOmega }(u_{\varepsilon }+1)^{M+1}+\int _{\varOmega } u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}(u_{\varepsilon }+1)v_\varepsilon ^{-\alpha }\\ =&\,\,\varepsilon \displaystyle \int _{\varOmega }(u_{\varepsilon }+1)^M A^{-1}(u_{\varepsilon }+1)+\int _{\varOmega }u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}v_\varepsilon ^{-\alpha } A^{-1}(u_{\varepsilon }+1)\\&+ \beta \int _{\varOmega } u_{\varepsilon } f(w_{\varepsilon })A^{-1}(u_{\varepsilon }+1). \end{aligned} \end{aligned}$$

(5.4.3)

Thanks to $W^{2,2}(\varOmega )\hookrightarrow L^{\infty }(\varOmega )$ in two-dimensional setting and the standard elliptic regularity in $L^2(\varOmega )$, one can find $C_1>0$ and $C_2>0$ such that

$$\begin{aligned} \Vert \varphi \Vert ^{M+1}_{L^{M+1}(\varOmega )}\le C_1\Vert \varphi \Vert ^{M+1}_{W^{2,2}(\varOmega )}\le C_2\Vert A\varphi \Vert ^{M+1}_{L^2(\varOmega )} \end{aligned}$$

for all $\varphi \in W^{2,2}(\varOmega )$ such that $\frac{\partial \varphi }{\partial \nu }|_{ \partial \varOmega }=0$. Hence, by the Young inequality, we can see that

$$\begin{aligned} \begin{aligned} \varepsilon \displaystyle \int _{\varOmega }(u_{\varepsilon }+1)^MA^{-1}(u_{\varepsilon }+1)\le&\displaystyle \frac{\varepsilon }{2}\int _{\varOmega }(u_{\varepsilon }+1)^{M+1}+ \displaystyle \frac{\varepsilon }{2}\int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{M+1}\\ \le&\,\, \displaystyle \frac{\varepsilon }{2} \Vert u_{\varepsilon } +1\Vert ^{M+1}_{L^{M+1}(\varOmega )}+ \displaystyle \frac{\varepsilon C_1}{2}\Vert A^{-1}(u_{\varepsilon }+1)\Vert ^{M+1}_{W^{2,2}(\varOmega )}\\ =&\,\,\displaystyle \frac{\varepsilon }{2} \displaystyle \int _{\varOmega }(u_{\varepsilon }+1)^{M+1}+\displaystyle \frac{\varepsilon C_1 C_2}{2}\Vert u_{\varepsilon }+1\Vert ^{M+1}_{L^2(\varOmega )}, \end{aligned} \end{aligned}$$

which along with the Young inequality implies that for any $\varepsilon _1>0$, there exists $c(\varepsilon _1)>0$ such that $\Vert \varphi \Vert _{L^2(\varOmega )}\le \varepsilon _1\Vert \varphi \Vert _{L^{M+1}(\varOmega )}+c(\varepsilon _1)\Vert \varphi \Vert _{L^1(\varOmega )}$ due to $M>1$ and entails that

$$\begin{aligned} \varepsilon \displaystyle \int _{\varOmega }(u_{\varepsilon }+1)^MA^{-1}(u_{\varepsilon }+1)\le \displaystyle \frac{3\varepsilon }{4} \displaystyle \int _{\varOmega }(u_{\varepsilon }+1)^{M+1}+C_3\Vert u_{\varepsilon }+1\Vert ^{M+1}_{L^1(\varOmega )}. \end{aligned}$$

Furthermore, since $\Vert w_\varepsilon (\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \Vert w_0\Vert _{L^\infty (\varOmega )}$, we apply Lemma 5.4 and Young’s inequality to obtain that for $t>0$,

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{\varOmega }u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1} v_\varepsilon ^{-\alpha } A^{-1}(u_{\varepsilon }+1) \\ \le&\,\,\displaystyle \frac{1}{4} \displaystyle \int _{\varOmega }\left\{ u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}\right\} ^{\frac{m+1}{m}} v_\varepsilon ^{-\alpha }+ C_4\int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1}v_\varepsilon ^{-\alpha }\\ \le&\displaystyle \frac{1}{4}\int _{\varOmega }u^{\frac{m+1}{m}}_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{\frac{m^2-1}{m}}v_\varepsilon ^{-\alpha } +C_4\delta ^{-\alpha } \int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1} \end{aligned} \end{aligned}$$

(5.4.4)

and

$$\begin{aligned} \begin{aligned}&\beta \displaystyle \int _{\varOmega } u_{\varepsilon }f(w_{\varepsilon })A^{-1}(u_{\varepsilon }+1)\\ \le&\,\, \displaystyle \frac{1}{4} \int _{\varOmega }u_{\varepsilon }^{m+1} v_\varepsilon ^{-\alpha } +C_5\int _{\varOmega }v_{\varepsilon }^{\frac{\alpha }{m}}|A^{-1}(u_{\varepsilon }+1)|^{\frac{m+1}{m}} \\ \le&\,\,\displaystyle \frac{1}{4}\int _{\varOmega }u^{m+1}_{\varepsilon } v_\varepsilon ^{-\alpha } + \int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1}+C_6\int _{\varOmega }v_{\varepsilon }^{\frac{\alpha }{m-1}}.\end{aligned} \end{aligned}$$

(5.4.5)

Noticing that $u_{\varepsilon }+1\ge \max \{u_{\varepsilon }+\varepsilon ,\varepsilon \}$, we have

$$ \int _{\varOmega }u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}(u_{\varepsilon }+1)v_\varepsilon ^{-\alpha }\ge \frac{1}{4}\int _{\varOmega }u_{\varepsilon }^{\frac{m+1}{m}}(u_{\varepsilon }+\varepsilon )^{\frac{m^2-1}{m}}v_\varepsilon ^{-\alpha }+ \frac{3}{4}\int _{\varOmega }u_{\varepsilon }^{m+1}v_\varepsilon ^{-\alpha }, $$

and hence insert (5.4.5) and (5.4.4) into (5.4.3) to get

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^2+ \int _{\varOmega }u_{\varepsilon }^{m+1}v_{\varepsilon }^{-\alpha } \\ \le&2(C_4\delta ^{-\alpha }+1)\displaystyle \int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1}+2C_6\int _{\varOmega }v_{\varepsilon }^{\frac{\alpha }{m-1}}, \end{aligned} \end{aligned}$$

which along with Lemma 5.5 and $D\ge 1$ readily arrive at (5.4.1).

By means of suitable interpolation arguments, one can appropriately estimate the integrals $\int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1}$ and $\int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^2$ in terms of $\int _{\varOmega }u_{\varepsilon }^{m+1}v_{\varepsilon }^{-\alpha }$ and thereby derive estimate of the form

$$ \int ^{t+1}_t\int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\le C $$

with $C>0$ independent of D, which can be stated as follows.

Lemma 5.9

Let $m>1$ and $D\ge 1$. Then there exists $C>0$ such that for all $D\ge 1$ as well as $\varepsilon \in (0,1)$

$$\begin{aligned} \int ^{t+1}_t\int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\le C~~~ ~~\hbox {for all}~~ t>0. \end{aligned}$$

(5.4.6)

Proof

By the standard elliptic regularity in $L^2(\varOmega )$, we have

$$\displaystyle \int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1}\le C_1\displaystyle \Vert u_{\varepsilon }+1\Vert _{L^2(\varOmega )}^{m+1}. $$

Noticing that for the given $p\in (2,m+1)$ (for example, $p:=\frac{m+3}{2}$), an application of Young’s inequality implies that for any $\eta >0$, there exists $C_1(\eta )>0$ such that

$$C_1\Vert u_{\varepsilon }+1\Vert ^{m+1}_{L^2(\varOmega )}\le \eta \Vert u_{\varepsilon }+1\Vert ^{m+1}_{L^p(\varOmega )} +C_1(\eta )\Vert u_{\varepsilon }+1\Vert ^{m+1}_{L^1(\varOmega )}.$$

On the other hand, by the Hölder inequality, we can see that

$$\begin{aligned} \int _{\varOmega }u^{p}_{\varepsilon }= & {} \,\,\int _{\varOmega }\left( u^{m+1}_{\varepsilon }v_{\varepsilon }^{-\alpha }\right) ^{\frac{p}{m+1}}v_{\varepsilon }^{\frac{p\alpha }{m+1}}\nonumber \\\le & {} \left( \int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\right) ^{\frac{p}{m+1}} \left( \int _{\varOmega }v^{\frac{p\alpha }{m+1-p}}_{\varepsilon }\right) ^{\frac{m+1-p}{m+1}}. \end{aligned}$$

Hence, combining the above estimates with Lemma 5.5, we arrive at

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\varOmega }|A^{-1}(u_{\varepsilon }+1)|^{m+1}&\le \eta \displaystyle \Vert u_{\varepsilon }+1\Vert _{L^p(\varOmega )}^{m+1} +C_1(\eta )\Vert u_{\varepsilon }+1\Vert ^{m+1}_{L^1(\varOmega )}\\&\le \eta \displaystyle \Vert u_{\varepsilon }\Vert _{L^p(\varOmega )}^{m+1}+C_2(\eta )\\&\le \displaystyle \eta \left( \int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\right) \left( \int _{\varOmega }v^{\frac{p\alpha }{m+1-p}}_{\varepsilon }\right) ^{\frac{m+1-p}{p}}+C_2(\eta )\\&\le \displaystyle \eta C_3(\alpha ,m)\left( \int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\right) +C_2(\eta )~~~\hbox {for all}~~t>0. \end{aligned} \end{aligned}$$

(5.4.7)

On the other hand, by self-adjointness of $A^{-\frac{1}{2}}$ and Hölder’s inequality, we get

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^{2}=&\,\, \displaystyle \int _{\varOmega }(u_{\varepsilon }+1) A^{-1}(u_{\varepsilon }+1)\\ \le&\,\, \displaystyle \Vert u_{\varepsilon }+1\Vert _{L^2(\varOmega )} \Vert A^{-1}(u_{\varepsilon }+1)\Vert _{L^2(\varOmega )}\\ \le&\,\, C_4\displaystyle \Vert u_{\varepsilon }+1\Vert ^2_{L^2(\varOmega )}\\ \le&\,\, C_5\displaystyle \Vert u_{\varepsilon }\Vert ^2_{L^{2}(\varOmega )}+C_5\\ \le&\,\, \displaystyle C_6\Vert u_{\varepsilon }\Vert ^{m+1}_{L^{2}(\varOmega )}+C_6~~~\hbox {for all}~~t>0. \end{aligned} \end{aligned}$$

So in this position, proceeding in the same way as above, we also have

$$\begin{aligned} \displaystyle \int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^{2}\le & {} C_6\eta \Vert u_{\varepsilon }+1\Vert ^{m+1}_{L^p(\varOmega )}+C_1(\eta )C_6\Vert u_{\varepsilon }+1\Vert ^{m+1}_{L^1(\varOmega )}\nonumber \\\le & {} \displaystyle \eta C_3(\alpha ,m)\left( \int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\right) +C_7(\eta )~~~\hbox {for all}~~t>0. \end{aligned}$$

(5.4.8)

Therefore, inserting (5.4.7) and (5.4.8) into (5.4.1) and taking $\eta$ sufficiently small, we have

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^2+ C_8\int _{\varOmega }|A^{-\frac{1}{2}}(u_{\varepsilon }+1)|^2 +C_8 \int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }\le C_9~~~\hbox {for all}~~t>0 \end{aligned}$$

with some $C_8>0, C_9>0$ for all $D\ge 1$. Furthermore, by Lemma 3.4 of Stinner et al. (2014), we immediately obtain (5.4.6).

As the direct consequence of Lemmas 5.9 and 5.5, we have the following.

Lemma 5.10

Let $m>1,D\ge 1$, then for $p\in (\max \{2,\frac{m+1}{\alpha +1}\}, m+1)$, one can find a constant $C(p)>0$ such that

$$\begin{aligned} \int ^{t+1}_{t}\int _{\varOmega }u^{p}_{\varepsilon }(\cdot ,s)ds\le C(p) ~ ~~\hbox {for all}~~ t>0~\hbox {and}~D\ge 1. \end{aligned}$$

(5.4.9)

Proof

For $p\in (2,m+1)$, we utilize Young’s inequality to estimate

$$\begin{aligned} \begin{aligned} \displaystyle \int ^{t+1}_{t}\int _{\varOmega }u^{p}_{\varepsilon }&=\displaystyle \int ^{t+1}_{t}\int _{\varOmega }\left( u^{m+1}_{\varepsilon }v_{\varepsilon }^{-\alpha }\right) ^{\frac{p}{m+1}}v_{\varepsilon }^{\frac{p\alpha }{m+1}} \\&\le \displaystyle \int ^{t+1}_{t}\int _{\varOmega }u^{m+1}_{\varepsilon }v^{-\alpha }_{\varepsilon }+\int ^{t+1}_{t}\int _{\varOmega }v^{\frac{p\alpha }{m+1-p}}_{\varepsilon } ~~~\hbox {for all}~~t>0, \end{aligned} \end{aligned}$$

which leads to (5.4.9) with the help of Lemma 5.5.

5.4.2 Boundedness of Solutions $(u_{\varepsilon }, v_{\varepsilon }, w_{\varepsilon })$

On the basis of the quite well-established arguments from parabolic regularity theory, we can turn the space–time integrability properties of $u^{p}_{\varepsilon }$ into the integrability properties of $\nabla v_{\varepsilon }$ as well as $\nabla w_{\varepsilon }$.

Lemma 5.11

Let $m>1,\alpha >0$ and suppose that $D\ge 1$. Then for $q\in (2,\frac{2(m+1)}{(3-m)_+})$, there exists constant $C>0$ such that for all $D\ge 1$ and $\varepsilon \in (0,1)$

$$\begin{aligned} \Vert v_{\varepsilon }(\cdot ,t)\Vert _{W^{1,q}(\varOmega )}\le C \end{aligned}$$

(5.4.10)

as well as

$$\begin{aligned} \Vert w_{\varepsilon }(\cdot ,t)\Vert _{W^{1,q}(\varOmega )}\le C \end{aligned}$$

(5.4.11)

for all $t> 0$.

Proof

From the continuity of function $h(x)=\frac{2x}{(4-x)_+}$ for $x\in [2,4)$, it follows that for given $q>2$ suitably close to the number $\frac{2(m+1)}{(3-m)_+}$, one can choose $p\in (2,m+1)$ in an appropriately small neighborhood of $m+1$ such that

$$\begin{aligned} \frac{p}{p-1}\cdot \left( \frac{1}{2}+\frac{1}{p}-\frac{1}{q}\right) <1. \end{aligned}$$

(5.4.12)

From the smoothing properties of Neumann heat semigroup $(e^{t\varDelta })_{t\ge 0}$, it follows that there exist $C_i>0 (i=1,2)$ such that

$$\begin{aligned} \Vert e^{\varDelta }\varphi \Vert _{W^{1,q}(\varOmega )}\le C_1\Vert \varphi \Vert _{L^1(\varOmega )}\quad \text {for }\varphi \in C^0(\overline{\varOmega }) \end{aligned}$$

(5.4.13)

as well as

$$\begin{aligned} \Vert e^{t\varDelta }\varphi \Vert _{W^{1,q}(\varOmega )}\le C_2 t^{-\frac{1}{2}-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})}\Vert \varphi \Vert _{L^p(\varOmega )} \quad \text {for all }t\in (0,1)\text { and }\varphi \in C^0(\overline{\varOmega }). \end{aligned}$$

(5.4.14)

Therefore, by the Duhamel representation to the second equation of (5.2.35), we obtain

$$\begin{aligned}&\Vert \tilde{v}_{\varepsilon }(\cdot ,t)\Vert _{W^{1,q}(\varOmega )} \\ =&\,\,\Vert e^{(t-(t-1)_+)(\varDelta -D^{-1})}\tilde{v}_{\varepsilon }(\cdot ,(t-1)_+)+ \displaystyle \frac{1}{D} \int ^t_{(t-1)_+}e^{(t-s) (\varDelta -D^{-1})}u_{\varepsilon }(\cdot , \frac{s}{D}) ds\Vert _{W^{1,q}(\varOmega )}\nonumber \\ \le&\,\, \Vert e^{(t-(t-1)_+)\varDelta }\tilde{v}_{\varepsilon }(\cdot ,(t-1)_+)\Vert _{W^{1,q}(\varOmega )}\!+\!\displaystyle \frac{1}{D} \int ^{t}_{(t-1)_+}\!\! \Vert e^{(t-s)\varDelta }u_{\varepsilon }(\cdot , \frac{s}{D})\Vert _{W^{1,q}(\varOmega )} d s. \nonumber \end{aligned}$$

(5.4.15)

Due to (5.4.13) and (5.4.14), we have

$$\begin{aligned} \begin{aligned} \Vert e^{(t-(t-1)_+)\varDelta }\tilde{v}_{\varepsilon }(\cdot ,(t-1)_+)\Vert _{W^{1,q}(\varOmega )}=&\,\, \Vert e^{\varDelta }\tilde{v}_{\varepsilon }(\cdot ,t-1)\Vert _{W^{1,q}(\varOmega )}\\ \le&\,\,C_1\Vert \tilde{v}_{\varepsilon }(\cdot ,t-1)\Vert _{L^1(\varOmega )} ~~\hbox {for}~ t>1, \end{aligned} \end{aligned}$$

(5.4.16)

while for $t\le 1$,

$$\begin{aligned} \begin{aligned} \Vert e^{(t-(t-1)_+)\varDelta }\tilde{v}_{\varepsilon }(\cdot ,(t-1)_+)\Vert _{W^{1,q}(\varOmega )}=&\,\, \Vert e^{t \varDelta }v_0(\cdot )\Vert _{W^{1,q}(\varOmega )}\\ \le&\,\,C_1\Vert v_0(\cdot )\Vert _{W^{1,\infty }(\varOmega )}. \end{aligned} \end{aligned}$$

On the other hand, we can see that for $t>0$

$$\begin{aligned} \begin{aligned}&\displaystyle \int ^{t}_{(t-1)_+}\Vert e^{-(t-s)\varDelta }u_{\varepsilon }(\cdot , D^{-1}s)\Vert _{W^{1,q}(\varOmega )}\\ \le&\,\, C_2\displaystyle \int ^{t}_{(t-1)_+}(t-s)^{-\frac{1}{2}-(\frac{1}{p}-\frac{1}{q})}\Vert u_{\varepsilon }(\cdot ,D^{-1}s)\Vert _{L^p(\varOmega )}ds\\ \le&\,\, C_2\displaystyle \left\{ \int ^{t}_{(t-1)_+}(t-s)^{^{-\frac{p}{p-1}(\frac{1}{2}+\frac{1}{p}-\frac{1}{q})}}ds\right\} ^{\frac{p-1}{p}} \left\{ \int ^{t}_{(t-1)_+}\Vert u_{\varepsilon }(\cdot ,D^{-1}s)\Vert _{L^p(\varOmega )}^{p}ds\right\} ^{\frac{1}{p}}\\ \le&\,\, C_2 \displaystyle (\int ^{1}_{0}\sigma ^{^{-\frac{p}{p-1}\cdot \left( \frac{1}{2}+\frac{1}{p}-\frac{1}{q}\right) }}d\sigma )^{\frac{p-1}{p}} \left\{ \int ^{t}_{(t-1)_+}\Vert u_{\varepsilon }(\cdot ,D^{-1}s)\Vert _{L^p(\varOmega )}^{p}ds\right\} ^{\frac{1}{p}}\\ \le&\,\, C_2 D^{\frac{1}{p}} \displaystyle (\int ^{1}_{0}\sigma ^{^{-\frac{p}{p-1}\cdot \left( \frac{1}{2}+\frac{1}{p}-\frac{1}{q}\right) }}d\sigma )^{\frac{p-1}{p}} \left\{ \int ^{D^{-1}t}_{(D^{-1}t-D^{-1})_+}\Vert u_{\varepsilon }(\cdot ,s)\Vert _{L^p(\varOmega )}^{p}ds \right\} ^{\frac{1}{p}}\\ \le&\,\, C_3 D^{\frac{1}{p}}, \end{aligned} \end{aligned}$$

(5.4.17)

where due to $D\ge 1$ and the application of Lemma 5.10, we have

$$\begin{aligned} \int ^{D^{-1}t}_{(D^{-1}t-D^{-1})_+}\Vert u_{\varepsilon }(\cdot ,s)\Vert _{L^p(\varOmega )}^{p}ds \le C_4 \end{aligned}$$

and the finiteness of $\int ^{1}_{0}\sigma ^{^{-\frac{p}{p-1}\cdot \left( \frac{1}{2}+\frac{1}{p}-\frac{1}{q}\right) }}d\sigma$ due to (5.4.12). Hence, combining (5.4.15) with (5.4.16) and (5.4.17) gives

$$\begin{aligned}\begin{aligned} \Vert v_{\varepsilon }(\cdot ,t)\Vert _{W^{1,q}(\varOmega )}&\le C_2\Vert \tilde{v}_{\varepsilon }(\cdot ,t-1)\Vert _{L^1(\varOmega )} +C_3 D^{\frac{1}{p}-1}+C_1\Vert v_0(\cdot )\Vert _{W^{1,\infty }(\varOmega )}\\&\le \displaystyle C_2(\int _{\varOmega }u_0+\beta \int _{\varOmega }w_0)+C_3+C_1\Vert v_0(\cdot )\Vert _{W^{1,\infty }(\varOmega )} \end{aligned} \end{aligned}$$

for all $t>0$ and thus completes the proof of (5.4.10).

Next due to $\Vert w_{\varepsilon }(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le \Vert w_{0}\Vert _{L^{\infty }(\varOmega )}$, an application of the Duhamel representation to the third equation in (5.2.20) yields

$$\begin{aligned} \Vert w_{\varepsilon }(\cdot ,t)\Vert _{W^{1,q}(\varOmega )}\le&\,\, \left\| e^{\varDelta }w_{\varepsilon }\left( \cdot ,(t-1)_+\right) \right\| _{W^{1,q}(\varOmega )} \\&+f(\Vert w_{0}\Vert _{L^{\infty }(\varOmega )})\int ^{t}_{(t-1)_+}\left\| e^{(t-s)\varDelta }u_{\varepsilon }(\cdot ,s)\right\| _{W^{1,q}(\varOmega )}ds, \end{aligned}$$

and thereby (5.4.11) can be actually derived as above.

The following lemma will be used in the derivation of regularity features about spatial and temporal derivatives of $u_{\varepsilon }$.

Lemma 5.12

Let $p>0$ and $\varphi \in C^{\infty }(\overline{\varOmega })$, then

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{p}\displaystyle \int _{\varOmega }\frac{d}{dt}(u_{\varepsilon }+\varepsilon )^p\cdot \varphi +(p-1)M\varepsilon \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p-2}(u_{\varepsilon }+1)^{M-1}|\nabla u_{\varepsilon }|^2\varphi \\ =&\,\,(1-p)\displaystyle \int _{\varOmega }(mu_{\varepsilon }+\varepsilon )(u_{\varepsilon }+\varepsilon )^{m+p-4}v^{-\alpha }_{\epsilon }|\nabla u_{\varepsilon }|^2\varphi \\&+\alpha (p-1)\displaystyle \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m+p-3}v_{\varepsilon }^{-\alpha -1}\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }\varphi \\&+(1-p)\displaystyle \int _{\varOmega }(mu_{\varepsilon }+\varepsilon )(u_{\varepsilon }+\varepsilon )^{m+p-3}v_{\varepsilon }^{-\alpha } \nabla u_{\varepsilon }\cdot \nabla \varphi \\&-M\varepsilon \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p-1}(u_{\varepsilon }+1)^{M-1}\nabla u_{\varepsilon }\cdot \nabla \varphi \\&+ \alpha \displaystyle \int _{\varOmega }u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m+p-2}v_{\varepsilon }^{-\alpha -1}\nabla v_{\varepsilon }\cdot \nabla \varphi +\beta \displaystyle \int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon })(u_{\varepsilon }+\varepsilon )^{p-1}\varphi \end{aligned} \end{aligned}$$

(5.4.18)

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

Proof

This can be verified by the straightforward computation.

Thanks to the boundedness of $\Vert \nabla v_{\varepsilon }(\cdot ,t)\Vert _{L^q(\varOmega )}$ with some $q>2$ in Lemma 5.11, we can achieve the following D-independent $L^{p}$-estimate of $u_{\varepsilon }$ with finite p.

Lemma 5.13

Let $m>1$. Then for all $D\ge 1$ and any $p>1$, there exists a constant $C(p)>0$ such that

$$\begin{aligned} \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^{p}(\varOmega )}\le C(p) \end{aligned}$$

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

Proof

According to Lemmas 5.4 and 5.5, one can find $C_i>0 (i=1,2)$ independent of $D\ge 1$ fulfilling

$$\begin{aligned} v_{\varepsilon }^{-\alpha }(x,t) \ge C_1, \quad v_{\varepsilon }^{-\alpha -2}(x,t)\le C_2 ~~\hbox {in}~~\varOmega \times (0,\infty ) \end{aligned}$$

(5.4.19)

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

Letting $\varphi \equiv 1$ in (5.4.18) and by Young’s inequality, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt} \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^p+ p(p-1)\displaystyle \int _{\varOmega }(mu_{\varepsilon }+\varepsilon )(u_{\varepsilon }+\varepsilon )^{m+p-4}v_{\varepsilon }^{-\alpha }|\nabla u_{\varepsilon }|^2 +\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p} \\ \le&\,\,\alpha p(p-1)\displaystyle \int _{\varOmega }u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m+p-3}v_{\varepsilon }^{-\alpha -1}\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon } \\&+\beta p \displaystyle \int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon })(u_{\varepsilon }+\varepsilon )^{p-1}+\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p} \\ \le&\,\,\displaystyle \frac{p(p-1)}{2} \displaystyle \int _{\varOmega }(mu_{\varepsilon }+\varepsilon )(u_{\varepsilon }+\varepsilon )^{m+p-4}v_{\varepsilon }^{-\alpha }|\nabla u_{\varepsilon }|^2 \\&+\frac{\alpha ^2p(p-1)}{2} \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m+p-1} v_{\varepsilon }^{-\alpha -2}|\nabla v_{\varepsilon }|^2 \\&+\beta pf (\Vert w_0\Vert _{L^{\infty }(\varOmega )}) \displaystyle \int _{\varOmega }u_{\varepsilon } (u_{\varepsilon }+\varepsilon )^{p-1}+\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p}. \end{aligned} \end{aligned}$$

Furthermore, recalling (5.4.19), we can find $C_3>0$ and $C_4>0$ independent of p such that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt} \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^p+C_3 \displaystyle \int _{\varOmega }|\nabla (u_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}|^2 +\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p}\\ \le&\,\, C_4p^2\displaystyle \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m+p-1} |\nabla v_{\varepsilon }|^2+ C_4 p \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p}. \end{aligned} \end{aligned}$$

(5.4.20)

According to (5.4.10), $\Vert \nabla v_{\varepsilon }\Vert ^2_{L^q(\varOmega )}\le C_5$ for any fixed $q\in (2,\frac{2(m+1)}{(3-m)_+})$, and hence, the Hölder inequality yields

$$\begin{aligned} \begin{aligned}&C_4p^2\displaystyle \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m+p-1} |\nabla v_{\varepsilon }|^2 \\ \le&C_4p^2\left\{ \displaystyle \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{\frac{(m+p-1)q}{q-2}} \right\} ^{1-\frac{2}{q}} \Vert \nabla v_{\varepsilon }\Vert ^2_{L^q(\varOmega )} \\ \le&C_4C_5p^2\Vert ( u_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^2_{L^{\frac{2q}{q-2}}(\varOmega )}\\ \le&\,\, \displaystyle \frac{C_3}{4}\displaystyle \int _{\varOmega }|\nabla (u_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}|^2+C_6(p), \end{aligned} \end{aligned}$$

(5.4.21)

where we have used an Ehrling-type inequality due to $W^{1,2}(\varOmega )\hookrightarrow L^{\frac{2q}{q-2}}(\varOmega )$ in two-dimensional setting and (5.2.28).

On the other hand, since

$$\begin{aligned}&C_4 p \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p} \\ \le&\,\, \eta \int _{\varOmega } (u_{\varepsilon }+\varepsilon )^{m+p-1}+ \frac{(C_4p)^{\frac{m-1+p}{m-1}}|\varOmega |}{\eta ^{\frac{p}{m-1}}}{=} \eta \Vert ( u_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}\Vert ^2_{L^2(\varOmega )}+ \frac{(C_4p)^{ \frac{m-1+p}{m-1} } |\varOmega |}{\eta ^{\frac{p}{m-1}}} \end{aligned}$$

for any $\eta >0$, we also have

$$\begin{aligned} C_4p\displaystyle \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p} \le \displaystyle \frac{C_3}{4}\displaystyle \int _{\varOmega }|\nabla (u_{\varepsilon }+\varepsilon )^{\frac{m+p-1}{2}}|^2+C_7(p) \end{aligned}$$

(5.4.22)

with some $C_7(p)>0$. Now inserting (5.4.22) and (5.4.21) into (5.4.20), we infer that for all $t>0$

$$\begin{aligned} \frac{d}{dt}\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{p}+\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^p\le C_8(p) \end{aligned}$$

with $C_8(p)>0$ independent of $D\ge 1$, which along with a standard comparison argument implies that

$$\begin{aligned} \int _{\varOmega }u^p_{\varepsilon }(\cdot ,t) \le \max \{C_8(p), \Vert u_0\Vert ^p_{L^P(\varOmega )}+1\} \end{aligned}$$

for all $t\ge 0$ and thus yields the claimed conclusion.

With the $L^{p}$-estimate of $u_{\varepsilon }$ at hand, the standard Moser-type iteration can be immediately applied in our approaches to obtain further regularity concerning $L^{\infty }$-norm of $u_{\varepsilon }$ (see Lemma A.1 of Tao and Winkler 2012a for example) and so we refrain from giving the details here.

Lemma 5.14

Assume that $m>1, \alpha >0$ and $D\ge 1$, then there exists $C>0$ such that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le C ~~\hbox {for all}~t\ge 0. \end{aligned}$$

(5.4.23)

Remark 5.3

It should be mentioned that when $m>1, \alpha >0$ and $D>0$, one can obtain the boundedness of $L^{\infty }$-norm of $u_{\varepsilon }$ for all $t>0$ by the above argument (also see Winkler 2020 for reference). However, the explicit dependence of $\Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^{p}(\varOmega )}$ on D is required to investigate the large time behavior of solutions in the sequel. Hence, $D\ge 1$ is imposed specially for the convenience of our discussion below.

At the end of this section, based on the above results, we derive a regularity property for v which goes beyond those in Lemma 5.11.

Lemma 5.15

Let $m>1,\alpha >0$. Then there exists $C>0$ such that for all $D\ge 1$ and $\varepsilon \in (0,1)$ such that

$$\begin{aligned} \Vert \nabla v_\varepsilon (\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le C \end{aligned}$$

(5.4.24)

as well as

$$\begin{aligned} \Vert \nabla w_\varepsilon (\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le C \end{aligned}$$

(5.4.25)

for all $t>D$.

Proof

Due to $\Vert \nabla e^{\varDelta }\tilde{v}(\cdot ,\tilde{t}-1) \Vert _{L^{\infty }(\varOmega )} \le C_1\Vert \tilde{v}(\cdot ,\tilde{t}-1)\Vert _{L^1(\varOmega )}$, as the proof of Lemma 5.11, we use the Duhamel formula of (5.2.35) in the following way:

$$\begin{aligned} \begin{aligned}&\quad \Vert \nabla \tilde{v}(\cdot ,\tilde{t})\Vert _{L^{\infty }(\varOmega )}\\&= \left\| \nabla e^{(\varDelta -D^{-1})}\tilde{v}(\cdot ,\tilde{t}-1)+D^{-1}\displaystyle \int ^{\tilde{t}}_{\tilde{t}-1}\nabla e^{(t-s) (\varDelta -D^{-1})}u(\cdot , D^{-1}s) ds\right\| _{L^{\infty }(\varOmega )}\\&\le \Vert \nabla e^{\varDelta }\tilde{v}(\cdot ,\tilde{t}-1)\Vert _{L^{\infty }(\varOmega )}+D^{-1} \displaystyle \int ^{\tilde{t}}_{\tilde{t}-1} \Vert \nabla e^{(\tilde{t}-s)\varDelta }u(\cdot , D^{-1}s)\Vert _{L^{\infty }(\varOmega )} d s\nonumber \\&\le C_1\Vert \tilde{v}(\cdot ,\tilde{t}-1)\Vert _{L^1(\varOmega )}+C_2D^{-1}\displaystyle \int ^{\tilde{t}}_{\tilde{t}-1} (1+(\tilde{t}-s)^{-\frac{3}{4}})ds \max _{\tilde{t}-1\le s\le \tilde{t}}\Vert u(\cdot ,D^{-1} s)\Vert _{L^{4}(\varOmega )}. \end{aligned} \end{aligned}$$

for all $\tilde{t}>1$, which along with (5.4.23) readily leads to (5.4.24). It is obvious that (5.4.25) can be proved similarly.

5.4.3 Asymptotic Behavior

1. Weak decay information

The standard parabolic regularity property becomes applicable to improve the regularity of u, v and w as follows.

Lemma 5.16

Let (u, v, w) be the nonnegative global solution of (5.1.18)–(5.1.20) obtained in Lemma 5.2. Then there exist $\kappa \in (0,1)$ and $C>0$ such that for all $t>D$

$$\begin{aligned} \Vert u\Vert _{C^{\kappa ,\frac{\kappa }{2}}(\overline{\varOmega }\times [t,t+1])}\le C \end{aligned}$$

(5.4.26)

as well as

$$\begin{aligned} \Vert v\Vert _{C^{2+\kappa ,1+\frac{\kappa }{2}}(\overline{\varOmega }\times [t,t+1])}+ \Vert w\Vert _{C^{2+\kappa ,1+\frac{\kappa }{2}}(\overline{\varOmega }\times [t,t+1])}\le C. \end{aligned}$$

(5.4.27)

Proof

We rewrite the first equation of (5.2.20) in the form

$$ u_{\varepsilon t}=\nabla \cdot a(x,t,u_{\varepsilon },\nabla u_{\varepsilon }) +b(x,t,u_{\varepsilon },\nabla u_{\varepsilon }) $$

where

$$ a(x,t,u_{\varepsilon },\nabla u_{\varepsilon })=(\varepsilon M (u_{\varepsilon }+1)^{M-1}+ mu_{\varepsilon }^{m-1}v_{\varepsilon }^{-\alpha }) \nabla u_{\varepsilon }-\alpha u_{\varepsilon }^m v_{\varepsilon }^{-\alpha -1}\nabla v_{\varepsilon }$$

and

$$b(x,t,u_{\varepsilon },\nabla u_{\varepsilon })=\beta u_{\varepsilon }f(w_{\varepsilon }).$$

According to Lemmas 5.4, 5.5, 5.14 and 5.15, there exist two constants $C_1>0$ and $C_2>0$ independent of $D\ge 1$ satisfying

$$\begin{aligned} C_1 \le v_{\varepsilon }^{-\alpha }(x,t) \le C_2 ~~\hbox {in}~~\varOmega \times (D,\infty ) \end{aligned}$$

and

$$ \Vert v_{\varepsilon }^{-\alpha -1}(\cdot ,t)\Vert _{L^\infty (\varOmega )}+ \Vert \nabla v_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )}+ \Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )} + \Vert w_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le C_2 $$

for $t\ge D$. This guarantees that for all $(x,t)\in \varOmega \times (D,\infty )$

$$ a(x,t,u_{\varepsilon },\nabla u_{\varepsilon })\cdot \nabla u_{\varepsilon }\ge \frac{C_1m}{2} u_{\varepsilon }^{m-1}|\nabla u_{\varepsilon }|^2-C_3, $$

$$ |a(x,t,u_{\varepsilon },\nabla u_{\varepsilon })|\le mC_4 u_{\varepsilon }^{m-1}|\nabla u_{\varepsilon }|+C_4|u_{\varepsilon }|^{\frac{m-1}{2}} $$

and

$$ |b(x,t,u_{\varepsilon },\nabla u_{\varepsilon })|\le C_5 $$

with some constants $C_i>0$ $(i=3,4,5)$ for all $t>D$ and $\varepsilon \in (0,1)$. Therefore, as an application of the known result on the Hölder regularity in scalar parabolic equations (Porzio and Vespri 1993), there exist $\kappa _1\in (0,1)$ and $C>0$ such that for all $t>D$ and $\varepsilon \in (0,1)$,

$$ \Vert u_{\varepsilon }\Vert _{C^{\kappa _1,\frac{\kappa _1}{2}}(\overline{\varOmega }\times [t,t+1])}\le C, $$

which along with (5.2.22) readily entails (5.4.26) with $\kappa =\kappa _1$. Similarly, one can also conclude that there exist $\kappa _2\in (0,1)$ and $C>0$ such that

$$\begin{aligned} \Vert v\Vert _{C^{\kappa _2,\frac{\kappa _2}{2}}(\overline{\varOmega }\times [t,t+1])}+ \Vert w\Vert _{C^{\kappa _2,\frac{\kappa _2}{2}}(\overline{\varOmega }\times [t,t+1])}\le C \quad \hbox {for all t>D}. \end{aligned}$$

Moreover, since $f\in C^1[0, \infty )$, we have

$$\Vert uf(w)\Vert _{C^{\kappa _3,\frac{\kappa _3}{2}}(\overline{\varOmega }\times [t,t+1])}\le C \quad \hbox {for all t>D} $$

with $\kappa _3=\min \{\kappa _1,\kappa _2\}$. Thereupon (5.4.27) with $\kappa =\kappa _3$ follows from the parabolic regularity estimates (Ladyzenskaja et al. 1968, Chap. IV, Theorem 5.3).

The first step toward establishing the stabilization result in Theorem 5.3 consists in the following observation.

Lemma 5.17

Assuming that $m>1$ and $D\ge 1$, we have

$$\begin{aligned} \int ^{\infty }_0\int _{\varOmega }uf(w)< \infty \end{aligned}$$

(5.4.28)

and

$$\begin{aligned} \int ^{\infty }_0\int _{\varOmega }|\nabla w|^2< \infty . \end{aligned}$$

(5.4.29)

Proof

An integration of the third equation in (5.1.18) yields

$$\begin{aligned} \int _{\varOmega }w_{\varepsilon }(\cdot ,t)+\int ^t_0\int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon })=\int _{\varOmega }w_0 \quad \hbox {for all}~~t>0. \end{aligned}$$

Since $w_{\varepsilon }\ge 0$, this entails

$$\begin{aligned} \int ^{\infty }_0\int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon }) \le \int _{\varOmega }w_0 \end{aligned}$$

(5.4.30)

which implies (5.4.28) on an application of Fatou’s lemma, because $u_{\varepsilon }f(w_{\varepsilon })\rightarrow uf(w)$ a.e. in $\varOmega \times (0,\infty )$.

We test the same equation by $w_{\varepsilon }$ to see that

$$\begin{aligned} \frac{1}{2}\int _{\varOmega }w^2_{\varepsilon }(\cdot ,t)+\int ^{t}_0\int _{\varOmega }|\nabla w_{\varepsilon }|^2=\frac{1}{2}\int _{\varOmega }w^2_0-\int ^t_0\int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon })w_{\varepsilon } \le \frac{1}{2}\int _{\varOmega }w^2_0 \end{aligned}$$

and thereby verifies (5.4.29) via (5.2.32).

The above decay information of $w_{\varepsilon }$ seems to be weak for the derivation of the large time behavior of $u_{\varepsilon }$ and $v_{\varepsilon }$. Indeed, under additional constraint on D, we obtain the decay information concerning the gradient of $u_{\varepsilon }$ and $v_{\varepsilon }$ which makes our latter analysis possible.

Lemma 5.18

Let $m>1$ and $\alpha >0$. There exists $D_0\ge 1$ such that whenever $D>D_0$, the solution of (5.1.18)–(5.1.20) constructed in Lemma 5.2 satisfies

$$\begin{aligned} \int ^{\infty }_{3}\int _{\varOmega }|\nabla u^{\frac{m+1}{2}} |^2 < \infty \end{aligned}$$

(5.4.31)

as well as

$$\begin{aligned} \int ^{\infty }_{3}\int _{\varOmega }|\nabla v |^2 < \infty . \end{aligned}$$

(5.4.32)

Proof

Testing the first equation of (5.2.20) by $(u_{\varepsilon }+\varepsilon )$ and applying Young’s inequality, we obtain that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{1}{2}\frac{d}{dt}\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^2+\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m-1} v^{-\alpha }_{\varepsilon }|\nabla u_{\varepsilon }|^2\\ \le&\displaystyle \alpha \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m}v^{-\alpha -1}_{\varepsilon }\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }+\beta \int _{\varOmega }u_{\varepsilon }(u_{\varepsilon }+\varepsilon )f(w_{\varepsilon })\\ \le&\displaystyle \frac{1}{2}\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m-1}v^{-\alpha }_{\varepsilon }|\nabla u_{\varepsilon }|^2 +\displaystyle \frac{\alpha ^2}{2} \int _{\varOmega }v_{\varepsilon }^{-\alpha -2} (u_{\varepsilon }+\varepsilon )^{m+1}|\nabla v_{\varepsilon }|^2 \\&+\beta \int _{\varOmega }(u_{\varepsilon }+\varepsilon )u_{\varepsilon } f(w_{\varepsilon }), \end{aligned} \end{aligned}$$

and hence

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{2}+\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m-1}v^{-\alpha }_{\varepsilon }|\nabla u_{\varepsilon }|^2\\ \le&\,\,\alpha ^2 \displaystyle \int _{\varOmega }v_{\varepsilon }^{-\alpha -2} (u_{\varepsilon }+\varepsilon )^{m+1}|\nabla v_{\varepsilon }|^2 +2\beta \int _{\varOmega }(u_{\varepsilon }+\varepsilon )u_{\varepsilon }f(w_{\varepsilon }). \end{aligned} \end{aligned}$$

(5.4.33)

On the other hand, let $\mu _{\varepsilon }(t)=\left( \frac{1}{|\varOmega |}\int _{\varOmega }u^{\frac{m+1}{2}}_{\varepsilon }(\cdot ,t)\right) ^{\frac{2}{m+1}}$, then testing the second equation of (5.2.20) by $-\varDelta v_{\varepsilon }$ shows

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }|\nabla v_{\varepsilon }|^2+2D\int _{\varOmega }(\varDelta v_{\varepsilon })^2+2\int _{\varOmega }|\nabla v_{\varepsilon }|^2\\ =&\,\,2\int _{\varOmega }(u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t))\varDelta v_{\varepsilon }\\ \le&\displaystyle \frac{1}{D}\int _{\varOmega }|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)|^2+D\int _{\varOmega }(\varDelta v_{\varepsilon })^2, \end{aligned} \end{aligned}$$

and thus

$$\begin{aligned} \displaystyle \frac{d}{dt}\int _{\varOmega }|\nabla v_{\varepsilon }|^2+D\int _{\varOmega }(\varDelta v_{\varepsilon })^2+2\int _{\varOmega }|\nabla v_{\varepsilon }|^2 \le \displaystyle \frac{1}{D}\int _{\varOmega }|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)|^2. \end{aligned}$$

(5.4.34)

Hence, combining (5.4.33) and (5.4.34), we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\left( \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^2+\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\right) +\eta D\int _{\varOmega }|\varDelta v_{\varepsilon }|^2+2\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\\&+\int _{\varOmega }v^{-\alpha }_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}|\nabla u_{\varepsilon }|^2\\ \le&\,\, \displaystyle \frac{\eta }{D}\int _{\varOmega }|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)|^2 +\alpha ^2 \int _{\varOmega }v_{\varepsilon }^{-\alpha -2} (u_{\varepsilon }+\varepsilon )^{m+1}|\nabla v_{\varepsilon }|^2 \\&+2\int _{\varOmega }(u_{\varepsilon }+\varepsilon )u_{\varepsilon }f(w_{\varepsilon }) \end{aligned} \end{aligned}$$

(5.4.35)

for parameter $\eta >0$ which will be determined later.

In view of Lemmas 5.4 and 5.5, there exist $C_i>0 (i=1,2)$ independent of $D\ge 1$ satisfying

$$\begin{aligned} v_{\varepsilon }^{-\alpha }(x,t) \ge C_1, \quad v_{\varepsilon }^{-\alpha -2}(x,t)\le C_2 ~~\hbox {in}~~\varOmega \times (2,\infty ) \end{aligned}$$

for all $\varepsilon \in (0,1)$. Therefore, from (5.4.35), it follows that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\left( \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^2+\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\right) +\eta D\int _{\varOmega }|\varDelta v_{\varepsilon }|^2+2\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\\&+C_1\int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m-1}|\nabla u_{\varepsilon }|^2\\ \le&\,\, \displaystyle \frac{\eta }{D}\int _{\varOmega }|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)|^2 +\alpha ^2 C_2 \int _{\varOmega } (u_{\varepsilon }+\varepsilon )^{m+1}|\nabla v_{\varepsilon }|^2 +2\int _{\varOmega }(u_{\varepsilon }+\varepsilon )u_{\varepsilon }f(w_{\varepsilon }). \end{aligned} \end{aligned}$$

(5.4.36)

According to Lemma 5.13 with $p=2(m+1)$, we have

$$ \displaystyle \left( \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{2(m+1)}\right) ^{\frac{1}{2}} \le C_3, $$

and then use the Gagliardo–Nirenberg inequality and the Hölder inequality to arrive at

$$\begin{aligned} \begin{aligned}&\displaystyle \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{m+1}|\nabla v_{\varepsilon }|^2\\ \le&\,\,\displaystyle \left( \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{2(m+1)}\right) ^{\frac{1}{2}}\left( \int _{\varOmega }|\nabla v_{\varepsilon }|^4\right) ^\frac{1}{2}\\ \le&\,\, C_4\displaystyle \left( \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^{2(m+1)}\right) ^{\frac{1}{2}}\left( \Vert \varDelta v_{\varepsilon }\Vert ^2 +\Vert \nabla v_{\varepsilon }\Vert ^2_{L^2(\varOmega )}\right) \\ \le&\,\, C_3C_4 (\Vert \varDelta v_{\varepsilon }\Vert ^2 +\Vert \nabla v_{\varepsilon }\Vert ^2_{L^2(\varOmega )}). \end{aligned} \end{aligned}$$

(5.4.37)

Therefore, inserting (5.4.37) into (5.4.36) yields

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\left( \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^2+\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\right) +\eta D\int _{\varOmega }|\varDelta v_{\varepsilon }|^2+2\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\\&+\frac{4C_1}{(m+1)^2}\int _{\varOmega } |\nabla (u_{\varepsilon }+\varepsilon )^{\frac{m+1}{2}}|^2\\ \le&\,\, \displaystyle \frac{\eta }{D}\int _{\varOmega }|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)|^2 +\alpha ^2 C_2 C_3C_4 (\Vert \varDelta v_{\varepsilon }\Vert ^2_{L^2(\varOmega )} +\Vert \nabla v_{\varepsilon }\Vert ^2_{L^2(\varOmega )}) \\&+2\int _{\varOmega }(u_{\varepsilon }+\varepsilon )u_{\varepsilon }f(w_{\varepsilon }). \end{aligned} \end{aligned}$$

(5.4.38)

By the elementary inequality:

$$ \frac{\xi ^\mu -\delta ^\mu }{\xi -\delta } \ge \delta ^{\mu -1}~~\hbox {for}~~\mu \ge 1,\xi \ge 0, \delta \ge 0~~\hbox { and}~~\xi \ne \delta , $$

we have

$$ |u_{\varepsilon }^{\frac{m+1}{2}}(\cdot ,t)-\mu _{\varepsilon }^{\frac{m+1}{2}}| \ge \mu _{\varepsilon }(\cdot ,t)^{\frac{m-1}{2}}|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)| $$

and thus

$$\begin{aligned} {\mu }^{m-1}_{\varepsilon }(t)\int _{\varOmega }|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)|^2\le \int _{\varOmega }|u^{\frac{m+1}{2}}_{\varepsilon }(\cdot ,t)-\mu ^{\frac{m+1}{2}}_{\varepsilon }(t)|^2. \end{aligned}$$

Furthermore, by the Hölder inequality and the noncreasing property of $t\mapsto \int _\varOmega u_\varepsilon (\cdot ,t)$,

$$\mu _\varepsilon (t)\ge \frac{1}{|\varOmega |}\int _\varOmega u_\varepsilon (\cdot ,t)\ge \frac{1}{|\varOmega |}\int _\varOmega u_0 $$

and thereby the Poincaré inequality entails that for some $C_5>0$

$$\begin{aligned} \begin{aligned}&\overline{u_0}^{m-1}\displaystyle \int _{\varOmega }|u_{\varepsilon }(\cdot ,t)-\mu _{\varepsilon }(t)|^2\\ \le&\,\,\displaystyle \int _{\varOmega }|u^{\frac{m+1}{2}}_{\varepsilon }(\cdot ,t)-\mu ^{\frac{m+1}{2}}_{\varepsilon }(t)|^2\\ \le&C_5\displaystyle \int _{\varOmega }|\nabla u^{\frac{m+1}{2}}_{\varepsilon }|^2\\ \le&C_5\displaystyle \int _{\varOmega }|\nabla (u_{\varepsilon }+\varepsilon )^{\frac{m+1}{2}}|^2. \end{aligned} \end{aligned}$$

(5.4.39)

Hence, substituting (5.4.39) into (5.4.38) shows that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\left( \int _{\varOmega }(u_{\varepsilon }+\varepsilon )^2+\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\right) + \left( \frac{4C_1}{(m+1)^2}-\frac{\eta C_5}{D\overline{u_0}^{m-1}})\int _{\varOmega } |\nabla (u_{\varepsilon }+\varepsilon \right) ^{\frac{m+1}{2}}|^2\\ \le&\,\, \displaystyle (\alpha ^2 C_2 C_3C_4-\eta D) \Vert \varDelta v_{\varepsilon }\Vert ^2_{L^2(\varOmega )} +(\alpha ^2 C_2 C_3C_4-2\eta )\Vert \nabla v_{\varepsilon }\Vert ^2_{L^2(\varOmega )}\\&+2\displaystyle \int _{\varOmega }(u_{\varepsilon }+\varepsilon )u_{\varepsilon }f(w_{\varepsilon })\\ \le&\,\, \displaystyle (\alpha ^2 C_2 C_3C_4-\eta ) \Vert \varDelta v_{\varepsilon }\Vert ^2_{L^2(\varOmega )} \\&+(\alpha ^2 C_2 C_3C_4-2\eta )\Vert \nabla v_{\varepsilon }\Vert ^2_{L^2(\varOmega )} +2\Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )}+1)\displaystyle \int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon }) \end{aligned} \end{aligned}$$

and hence completes the proof upon the choice of $D_0:=\max \{1,\frac{\alpha ^2 C_2 C_3C_4C_5(m+1)^2}{3C_1\overline{u_0}^{m-1}}\}$. Indeed, for any $D>D_0$, it is possible to find $\eta >0$ such that

$$\frac{3C_1}{(m+1)^2}\ge \frac{\eta C_5}{D\overline{u_0}^{m-1}},~~ \alpha ^2 C_2 C_3C_4\le \eta $$

and thereby

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\left( \int _{\varOmega }|u_{\varepsilon }+\varepsilon |^2+\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\right) + \frac{C_1}{(m+1)^2}\int _{\varOmega } |\nabla (u_{\varepsilon }+\varepsilon )^{\frac{m+1}{2}}|^2 +\eta \int _{\varOmega }|\nabla v_{\varepsilon }|^2\\ \le&\,\, 2(\Vert u_{\varepsilon }(\cdot ,t)\Vert _{L^\infty (\varOmega )}+1)\displaystyle \int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon }). \end{aligned} \end{aligned}$$

Therefore, in view of (5.4.30), (5.4.23) and (5.4.24), we see that for any $t>3$,

$$\begin{aligned} \displaystyle \int ^t_3\int _{\varOmega } |\nabla (u_{\varepsilon }+\varepsilon )^{\frac{m+1}{2}}|^2 +\displaystyle \int ^t_3\int _{\varOmega }|\nabla v_{\varepsilon }|^2 \le C_6+ C_6\displaystyle \int ^\infty _3\displaystyle \int _{\varOmega }u_{\varepsilon }f(w_{\varepsilon })\le C_6+C_6\int _\varOmega w_0 \end{aligned}$$

(5.4.40)

with constant $C_6>0$ independent of $\varepsilon$ and time t, which implies that (5.4.31) and (5.4.32) are valid due to the lower semi-continuity of norms.

2. Decay of w

The integrability statement in Lemma 5.17 can be turned into the decay property of w with respect to the norm in $L^\infty (\varOmega )$, thanks to the fact that $\Vert u(\cdot ,t)\Vert _{L^1(\varOmega )}$ is increasing with time, while $\Vert w(\cdot ,t)\Vert _{L^\infty (\varOmega )}$ is non-increasing.

Lemma 5.19

The third component of the weak solution of (5.1.18)–(5.1.20) constructed in Lemma 5.2 fulfills

$$\begin{aligned} \Vert w(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\rightarrow 0 \quad \text {as }t\rightarrow \infty . \end{aligned}$$

(5.4.41)

Proof

Writing $\overline{u_0}:=\frac{1}{|\varOmega |}\int _{\varOmega }u_0$ and $\overline{f(w)}:=\frac{1}{|\varOmega |}\int _{\varOmega }f(w)$, we use the Cauchy–Schwarz inequality and the Poincaré inequality to see that for all $t>0$

$$\begin{aligned} \overline{u_0}\cdot \int _{\varOmega }f(w)= & {} \int _{\varOmega }u\overline{f(w)}\\= & {} \int _{\varOmega }u f(w)-\int _{\varOmega }u(f(w)-\overline{f(w)})\\\le & {} \int _{\varOmega }u f(w)+C_1\Vert u\Vert _{L^{\infty }(\varOmega )} \Vert f'(w)\Vert _{L^{\infty }(\varOmega )}\left\{ \int _{\varOmega }|\nabla w|^2\right\} ^{\frac{1}{2}}. \end{aligned}$$

Thanks to the boundedness of u and w, we have

$$\begin{aligned} {\overline{u_0}}^2\cdot \left\{ \int _{\varOmega }f(w)\right\} ^{2}\le & {} 2\left\{ \int _{\varOmega }u f(w)\right\} ^2+C_2\int _{\varOmega }|\nabla w|^2\\\le & {} C_3\int _{\varOmega }u f(w)+C_2\int _{\varOmega }|\nabla w|^2. \end{aligned}$$

Hence, from Lemma 5.17, it follows that

$$\begin{aligned} \int ^{\infty }_1\Vert f(w(\cdot ,t))\Vert ^2_{L^1(\varOmega )}dt< \infty , \end{aligned}$$

which, along with the uniform Hölder estimate from Lemma 5.16, implies that

$$\begin{aligned} f(w(\cdot ,t))\rightarrow 0 \quad \text {in }L^1(\varOmega ) \quad \text {as }t\rightarrow \infty \end{aligned}$$

and thereby we may extract a subsequence $(t_j)_{j\in \mathbb {N}} \subset \mathbb {N}$ such that as $t_j\rightarrow \infty$, $f(w(\cdot ,t_j))\rightarrow 0$ almost everywhere in $\varOmega$. Recalling function f is positive on $(0,\infty )$ and $f(0)=0$, this necessarily requires that $w(\cdot ,t_j)\rightarrow 0$ almost everywhere in $\varOmega$ as $t_j\rightarrow \infty .$ Furthermore, the dominated convergence theorem ensures that

$$\begin{aligned} w(\cdot ,t_j)\rightarrow 0 \quad \text {in }L^1(\varOmega ) \quad \text {as }t_j\rightarrow \infty . \end{aligned}$$

Now invoking the Gagliardo–Nirenberg inequality in two dimensional setting, we have

$$\begin{aligned} \Vert w(\cdot ,t_j)\Vert _{L^{\infty }(\varOmega )}\le C_4\Vert \nabla w(\cdot ,t_j)\Vert ^{\frac{4}{5}}_{L^{4}(\varOmega )}\Vert w(\cdot ,t_j)\Vert ^{\frac{1}{5}}_{L^{1}(\varOmega )}+C_4\Vert w(\cdot ,t_j)\Vert _{L^1(\varOmega )} \end{aligned}$$

and thus

$$\begin{aligned} \Vert w(\cdot ,t_j)\Vert _{L^{\infty }(\varOmega )}\rightarrow 0 \quad \text {as }t_j\rightarrow \infty . \end{aligned}$$

(5.4.42)

Since $t\mapsto \Vert w(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}$ is noncreasing by Lemma 5.4, (5.4.41) indeed results from (5.4.42).

3. Convergence of u

Now we will show that u stabilizes toward the constant $\overline{u_0}+\beta \overline{w_0}$ as $t\rightarrow \infty$. Note that a first step in this direction is provided by the finiteness of $\int ^{\infty }_{3}\int _{\varOmega }|\nabla u^{\frac{m+1}{2}}|^2$ in Lemma 5.18, which implies that $\Vert \nabla u^{\frac{m+1}{2}}(\cdot ,t_k)\Vert _{L^2(\varOmega )}$ along a suitable sequence of numbers $t_k\rightarrow \infty$. However, in order to make sure convergence along the entire net $t\rightarrow \infty$, a certain decay property of $u_t$ seems to be required.

Lemma 5.20

We have

$$\begin{aligned} \int ^\infty _3\Vert u_t(\cdot ,t) \Vert ^2_{(W_0^{1,2}(\varOmega ))^*}dt <\infty . \end{aligned}$$

(5.4.43)

Proof

For any $\varphi \in C_0^{\infty }(\varOmega )$, multiplying the first equation in (5.2.20) by $\varphi$ and integrating by parts over $\varOmega$ yield

$$\begin{aligned} \begin{aligned}&\quad |\displaystyle \int _\varOmega u_{\varepsilon t}\varphi | \\&= \left| \displaystyle \int _\varOmega \varepsilon \nabla (u_{\varepsilon }+1)^{M}\cdot \nabla \varphi +\nabla (u_{\varepsilon }(u_{\varepsilon }+\varepsilon )^{m-1}v_{\varepsilon }^{-\alpha })\cdot \nabla \varphi + \beta u_{\varepsilon }f(w_{\varepsilon }) \varphi \right| \\&\le \displaystyle \int _\varOmega ( M(u_{\varepsilon }+1)^{M-1}|\nabla u_{\varepsilon }|+ mv_{\varepsilon }^{-\alpha }(u_{\varepsilon }+\varepsilon )^{m-1}|\nabla u_{\varepsilon }|+\alpha (u_{\varepsilon }+1)^m v_{\varepsilon }^{-\alpha -1}|\nabla v_{\varepsilon }|)|\nabla \varphi | \\&\quad +\displaystyle \beta \int _{\varOmega }|u_{\varepsilon }f(w_{\varepsilon })|\Vert \varphi \Vert _{L^\infty (\varOmega )}\\&\le C_1 (\left\{ \displaystyle \int _{\varOmega } |\nabla ( u_\varepsilon +\varepsilon )^{\frac{m+1}{2}}|^2\right\} ^{\frac{1}{2}}+\displaystyle \left\{ \int _{\varOmega } |\nabla v_\varepsilon |^2\right\} ^{\frac{1}{2}})\Vert \varphi \Vert _{W^{1,2}(\varOmega )}\\&\quad + \beta \displaystyle \int _{\varOmega }u_\varepsilon f(w_\varepsilon )\Vert \varphi \Vert _{L^\infty (\varOmega )} \end{aligned} \end{aligned}$$

with $C_1>0$ independent of $\varphi$ and $\varepsilon$, where we have used the boundedness of $u_\varepsilon$ and $v_\varepsilon$.

As in the considered two-dimensional setting we have $W^{1,2}(\varOmega )\hookrightarrow L^\infty (\varOmega )$, the above inequality implies that

$$ \Vert u_{\varepsilon t}(\cdot ,t) \Vert _{(W_0^{1,2}(\varOmega ))^*}\le C_1 ( \left\{ \displaystyle \int _{\varOmega } |\nabla (u_\varepsilon +\varepsilon )^{\frac{m+1}{2}}|^2\right\} ^{\frac{1}{2}}+\displaystyle \left\{ \int _{\varOmega } |\nabla v_\varepsilon |^2\right\} ^{\frac{1}{2}}) +\beta \displaystyle \int _{\varOmega }u_\varepsilon f(w_\varepsilon ) $$

for all $t> 3$ and hence for all $T> 4$,

$$\begin{aligned}&\displaystyle \int ^T_3\Vert u_{\varepsilon t}(\cdot ,t) \Vert ^2_{(W^{1,2}(\varOmega ))^*}dt \\ \le&\,\, C_2\left( \int ^T_3\displaystyle \int _{\varOmega } |\nabla (u_\varepsilon +\varepsilon )^{\frac{m+1}{2}}|^2 + \displaystyle \int ^T_3\int _{\varOmega } |\nabla v_\varepsilon |^2+ \displaystyle \int ^T_3\int _{\varOmega }u_\varepsilon f(w_\varepsilon )\right) \end{aligned}$$

which together with (5.4.40) leads to

$$ \int ^\infty _3\Vert u_{\varepsilon t}(\cdot ,t) \Vert ^2_{(W_0^{1,2}(\varOmega ))^*}dt \le C_3 $$

with $C_3>0$ independent of $\varepsilon$. Hence, (5.4.43) results from lower semi-continuity of the norm in the Hilbert space $L^2((3,\infty );(W_0^{1,2}(\varOmega ))^*)$ with respect to weak convergence.

Thanks to the above estimates, we adapt the argument in Winkler (2015b) to show that u actually stabilizes toward $\overline{u_0}+\beta \overline{w_0}$ in the claimed sense beyond in the weak-$*$ sense in $L^\infty (\varOmega )$.

Lemma 5.21

Let $m>1,\alpha >0$ and suppose that $D\ge D_0$ with $D_0$ as in Lemma 5.18. Then we have

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

(5.4.44)

where $u_{\star }=\frac{1}{|\varOmega |}\int _{\varOmega } u_0+\frac{\beta }{|\varOmega |}\int _{\varOmega }w_0$.

Proof

According to Lemmas 5.18 and 5.20, one can conclude that

$$\begin{aligned} u(\cdot ,t){\mathop {\rightharpoonup }\limits ^{\mathrm {w}^*}} u_{\star }\quad \text{ in }~ L^\infty (\varOmega )~~\text {as }~~t\rightarrow \infty . \end{aligned}$$

(5.4.45)

In fact, if this conclusion does not hold, then one can find a sequence $(t_k)_{k\in \mathbb {N}} \subset (0, \infty )$ such that $t_k \rightarrow \infty$ as $k \rightarrow \infty$, and some $\tilde{\psi }\in L^1(\varOmega )$ such that

$$ \int _\varOmega u(x, t_k)\tilde{\psi }dx-\int _\varOmega u_{\star }\tilde{\psi }dx\ge C_1 ~\hbox {for all}~k\in \mathbb {N} $$

with some $C_1 > 0$. Furthermore, by the boundedness of u and the density of $C_0^\infty (\varOmega )$ in $L^1(\varOmega )$, we can choose $\psi \in C_0^\infty (\varOmega )$ closing $\tilde{\psi }$ in $L^1(\varOmega )$ enough that

$$\begin{aligned} \int _\varOmega u(x, t_k)\psi dx-\int _\varOmega u_{\star }\psi dx\ge \frac{3C_1}{4} ~~~\hbox {for all}~k\in \mathbb {N} \end{aligned}$$

and then

$$\begin{aligned} \displaystyle \int ^{t_k+1}_{t_k}\int _\varOmega u(x, t)\psi dxdt-\displaystyle \int ^{t_k+1}_{t_k}\int _\varOmega u_{\star }\psi dxdt\ge \frac{C_1}{2} ~~~\hbox {for all sufficently large }~k\in \mathbb {N}, \end{aligned}$$

(5.4.46)

where we have used the fact that

$$\begin{aligned} \begin{aligned}&\left| \displaystyle \int ^{t_k+1}_{t_k}\int _\varOmega ( u(x, t)-u(x, t_k))\psi dx\right| \\ =&\,\,\left| \displaystyle \int ^{t_k+1}_{t_k}\int ^t_{t_k}\langle u_t(\cdot ,s), \psi (\cdot )\rangle ds dt\right| \\ \le&\,\, \displaystyle \int ^{t_k+1}_{t_k}\int ^t_{t_k}\Vert u_t(\cdot ,s)\Vert _{(W^{1,2}_0(\varOmega ))^*} ds dt\cdot \Vert \psi \Vert _{W^{1,2}_0(\varOmega )} \\[4mm] \le&\,\, \displaystyle \int ^{t_k+1}_{t_k}\left\{ \int ^t_{t_k}\Vert u_t(\cdot ,s)\Vert ^2_{(W^{1,2}_0(\varOmega ))^*}ds\right\} ^{\frac{1}{2}}|t-t_k|^{\frac{1}{2}} dt\cdot \Vert \psi \Vert _{W^{1,2}_0(\varOmega )} \\[4mm] \le&\,\, \left\{ \displaystyle \int ^{t_k+1}_{t_k}\int ^t_{t_k}\Vert u_t(\cdot ,s)\Vert ^2_{(W^{1,2}_0(\varOmega ))^*}ds dt \right\} ^{\frac{1}{2}}\cdot \Vert \psi \Vert _{W^{1,2}_0(\varOmega )} \\[4mm] \le&\,\, \left\{ \displaystyle \int ^\infty _{t_k}\Vert u_t(\cdot ,s)\Vert ^2_{(W^{1,2}_0(\varOmega ))^*}ds \right\} ^{\frac{1}{2}}\cdot \Vert \psi \Vert _{W^{1,2}_0(\varOmega )}\\ \longrightarrow&0~~\hbox {as}~~k \rightarrow \infty , \end{aligned} \end{aligned}$$

due to Lemma 5.20.

Let $\mu (t)=\left( \frac{1}{|\varOmega |}\int _{\varOmega }u^{\frac{m+1}{2}}(\cdot ,t)\right) ^{\frac{2}{m+1}}$. Then as in (5.4.39), we have

$$\begin{aligned} \overline{u_0}^{m-1}\int _{\varOmega }|u(\cdot ,t)-\mu (t)|^2\le \int _{\varOmega }|u^{\frac{m+1}{2}}(\cdot ,t)-\mu ^{\frac{m+1}{2}}(t)|^2\le C_5\int _{\varOmega }|\nabla u^{\frac{m+1}{2}}|^2 \end{aligned}$$

and thus

$$\begin{aligned} \overline{u_0}^{m-1}\displaystyle \int ^{t_k+1}_{t_k}\int _{\varOmega }|u(\cdot ,t)-\mu (t)|^2\le C_5\displaystyle \int ^{t_k+1}_{t_k}\int _{\varOmega }|\nabla u^{\frac{m+1}{2}}(\cdot ,t)|^2. \end{aligned}$$

(5.4.47)

We now introduce

$$ u_k(x, s) := u(x, t_k + s), (x, s)\in \varOmega \times (0, 1) $$

and

$$ \mu _k(x, s) := \mu (x, t_k + s), (x, s)\in \varOmega \times (0, 1) $$

for $k\in \mathbb {N}$. Then (5.4.47) implies that

$$\begin{aligned} \begin{aligned} \overline{u_0}^{m-1}\displaystyle \int ^{1}_{0}\int _{\varOmega }|u_k(\cdot ,s)-\mu _k(s)|^2ds\le&C_5\displaystyle \int ^{t_k+1}_{t_k}\int _{\varOmega }|\nabla u^{\frac{m+1}{2}}(\cdot ,t)|^2\\&\rightarrow 0~~\hbox {as }~~k\rightarrow \infty , \end{aligned} \end{aligned}$$

due to (5.4.31) in Lemma 5.18. This means that

$$\begin{aligned} u_k(x,s)-\mu _k(s)\rightarrow 0~~\hbox {in} ~~L^2(\varOmega \times (0,1))~~\hbox {as}~~k\rightarrow \infty , \end{aligned}$$

which in particular allows us to get

$$\begin{aligned} \displaystyle \int ^{1}_{0}\int _{\varOmega }(u_k(\cdot ,s)-\mu _k(s)) \psi ds\rightarrow 0~ \hbox {as}~~k\rightarrow \infty \end{aligned}$$

(5.4.48)

as well as

$$\begin{aligned} \displaystyle \int ^{1}_{0}\int _{\varOmega }(u_k(\cdot ,s)-\mu _k(s)) ds\rightarrow 0~ \hbox {as}~~k\rightarrow \infty . \end{aligned}$$

(5.4.49)

Moreover, by Lemma 5.19, we have

$$\begin{aligned} \displaystyle \int ^{t_k+1}_{t_k}\int _{\varOmega }w(\cdot ,t)dt\le |\varOmega |\Vert w(\cdot ,t_k)\Vert _{L^\infty (\varOmega )} \rightarrow 0~ \hbox {as}~~k\rightarrow \infty \end{aligned}$$

and thereby

$$\begin{aligned} |\varOmega |\displaystyle \int ^{1}_{0} \mu _k(s) ds=&\,\,\displaystyle \int ^{1}_{0}\int _{\varOmega }u_k(\cdot ,s)ds- \displaystyle \int ^{1}_{0}\int _{\varOmega }(u_k(\cdot ,s)-\mu _k(s)) ds \nonumber \\ =&\,\, |\varOmega |u_*-\beta \displaystyle \int ^{t_k+1}_{t_k}\int _{\varOmega }w(\cdot ,t)dt - \displaystyle \int ^{1}_{0}\int _{\varOmega }(u_k(\cdot ,s)-\mu _k(s)) ds \nonumber \\ \rightarrow&|\varOmega |u_* ~ \hbox {as}~~k\rightarrow \infty \end{aligned}$$

(5.4.50)

due to (5.4.49) and (5.4.41).

Therefore, from (5.4.46), (5.4.48) and (5.4.50), it follows that

$$\begin{aligned} \begin{aligned} \displaystyle \frac{C_1}{2} \le&\displaystyle \int ^{t_k+1}_{t_k}\int _\varOmega u(\cdot , t)\psi dt-\displaystyle \int ^{t_k+1}_{t_k}\int _\varOmega u_{\star }\psi dt \\[4mm] =&\,\, \displaystyle \int ^{1}_{0}\int _\varOmega (u_k(\cdot , s)-\mu _k(s)) \psi ds+\int ^{1}_{0}\int _\varOmega \mu _k(s) \psi ds -\displaystyle u_{\star }\int _\varOmega \psi \\ =&\,\, \displaystyle \int ^{1}_{0}\int _\varOmega (u_k(\cdot , s)-\mu _k(s)) \psi ds+\int ^{1}_{0} \mu _k(s) ds\int _\varOmega \psi -u_{\star } \displaystyle \int _\varOmega \psi \\ \rightarrow&0~ \hbox {as}~~k\rightarrow \infty , \end{aligned} \end{aligned}$$

which is absurd and hence proves that actually (5.4.45) is valid.

Let us suppose on the contrary that (5.4.44) be false. Then without loss of generality, there exist sequence $\{x_k\}_{k\in \mathbb {N}}$ and $\{t_k\}_{k\in \mathbb {N}}\in (0,\infty )$ with $t_k\rightarrow \infty$ as $k\rightarrow \infty$ such that for some $C_1>0$

$$ u(x_k,t_k)-u_*=\max _{x\in \varOmega }|u(x,t_k)-u_*|\ge C_1~~\hbox {for all} ~k\in \mathbb {N}. $$

In view of the compactness of $\overline{\varOmega }$, where passing to subsequences we can find $x_0\in \overline{\varOmega }$ such that $x_k\rightarrow x_0$ as $k\rightarrow \infty$. Furthermore, because u is uniformly continuous in $\bigcup _{~k\in \mathbb {N}}(\overline{\varOmega }\times t_k)$, this entails that one can extract a further subsequence if necessary such that

$$ u(x,t_k)-u_*\ge \frac{C_1}{2} ~~~\hbox {for all}~ x\in B:=B_{\delta }(x_0)\cap \varOmega ~ \hbox {and}~k\in \mathbb {N} $$

for some $\delta >0$. Noticing that if $x_0\in \partial \varOmega$, the smoothness of $\partial \varOmega$ ensures the existence of $\hat{x}_0\in \varOmega$ and a smaller $\hat{\delta }>0$ such that $B_{\hat{\delta }}(\hat{x}_0)\subset B$. Now taking the nonnegative function $\psi \in C^\infty _0(B_{\hat{\delta }}(\hat{x}_0)))$ such as a smooth truncated function in $B_{\hat{\delta }}(\hat{x}_0))$, we then have

$$ \int _\varOmega (u(x, t_k)-u_{\star } )\psi dx=\int _{B_{\hat{\delta }}(\hat{x}_0)}(u(x, t_k)-u_{\star } )\psi dx\ge \frac{C_1}{2}\cdot \int _\varOmega \psi dx, $$

which contradicts (5.4.45) and hence proves the lemma.

4. Stabilization of v

In what follows, based on the uniform Hölder bounds of v and decay of $\nabla v$ implied by (5.4.27) and (5.4.32), respectively, we shall show the corresponding stabilization result for v by a contradiction argument.

Lemma 5.22

Let $m>1$ and (u, v, w) be the solution of (5.1.18)–(5.1.20) obtained in Lemma 5.2. Then we have

$$\begin{aligned} \Vert v(\cdot ,t)-u_{\star }\Vert _{L^{\infty }(\varOmega )}\rightarrow 0\quad \text {as }t\rightarrow \infty . \end{aligned}$$

(5.4.51)

Proof

According to the uniform Hölder bounds of v and decay of $\nabla v$ implied by (5.4.27) and (5.4.32), respectively, (5.4.51) may be derived by a contradiction argument. Indeed, assume that (5.4.51) was false, then we can find a sequence $(t_k)_{k\in \mathbb {N}}$ with $t_k\rightarrow \infty$ as $k\rightarrow \infty$, and constant $C_1>0$ such that

$$\begin{aligned} \Vert v(\cdot ,t_{k})-u_{\star }\Vert _{L^{\infty }}\ge C_1. \end{aligned}$$

Furthermore, the uniform Hölder continuity of v in $\varOmega \times [t,t+1]$ warrants the existence of $(x_k)_{k\in \mathbb {N}}$ and $r>0$ such that

$$\begin{aligned} |v(x,t)-u_{\star }|>\frac{C_1}{2} \end{aligned}$$

for every $x\in B_{r}(x_k)$ and $t\in (t_k,t_k+\tau )$ and hence

$$\begin{aligned} \int ^{t_k+\tau }_{t_{k}}\int _{\varOmega }|v(\cdot ,t)-u_{\star }|^2>\frac{|\varOmega |\tau c^2_1}{4}. \end{aligned}$$

(5.4.52)

On the other hand, the Poincaré inequality indicates

$$\begin{aligned} \int ^{t_k+\tau }_{t_k}\int _{\varOmega }|v(\cdot ,t)-u_{\star }|^2\le C\int ^{t_k+\tau }_{t_k}\int _{\varOmega }|\nabla v|^2+C \int ^{t_k+\tau }_{t_k}\int _{\varOmega }|\overline{v(\cdot ,t)}-u_{\star }|^2. \end{aligned}$$

(5.4.53)

Therefore, (5.4.53) yields a contradiction to (5.4.52) thanks to

$$ \int ^{t_k+\tau }_{t_k}\int _{\varOmega }|\nabla v|^2\rightarrow 0 \quad \text {as }t_k\rightarrow \infty . $$

Now the convergence result in the flavor of Theorem 5.3 has actually been proved already.

Proof of Theorem 5.3.  The claimed assertion in Theorem 5.3 is the consequence of Lemmas 5.19, 5.21 and 5.22.