# Asymptotic behavior of a quasilinear Keller–Segel system with signal-
suppressed motility

Chi Xu1, Yifu Wang1,∗
School of Mathematics and Statistics, Beijing Institute of Technology
Beijing 100081, P.R. China1

###### Abstract

This paper is concerned with the density-suppressed motility model:
$u_{t}=\Delta(\displaystyle\frac{u^{m}}{v^{\alpha}})+\beta uf(w),v_{t}=D\Delta
v-v+u,w_{t}=\Delta w-uf(w)$ in a smoothly bounded convex domain
$\Omega\subset{\mathbb{R}}^{2}$, where $m>1$, $\alpha>0,\beta>0$ and $D>0$ are
parameters, the response function $f$ satisfies $f\in
C^{1}([0,\infty)),f(0)=0,f(w)>0$ in $(0,\infty)$. This system describes the
density-suppressed motility of Eeshcrichia coli cells in process of spatio-
temporal pattern formation via so-called self-trapping mechanisms. Based on
the duality argument, it is shown that for suitable large $D$ the problem
admits at least one global weak solution $(u,v,w)$ which will asymptotically
converge to the spatially uniform equilibrium
$(\overline{u_{0}}+\beta\overline{w_{0}},\overline{u_{0}}+\beta\overline{w_{0}},0)$
with $\overline{u_{0}}=\frac{1}{|\Omega|}\int_{\Omega}u(x,0)dx$ and
$\overline{w_{0}}=\frac{1}{|\Omega|}\int_{\Omega}w(x,0)dx$ in
$L^{\infty}(\Omega)$.

††∗Corresponding author. E-mail<EMAIL_ADDRESS>(Y.Wang),
<EMAIL_ADDRESS>(C. Xu)

Keywords: Signal-suppressed motility, Keller–Segel system, asymptotic
behavior.

MSC: 35B40; 35K57; 35Q92; 92C17.

##  1 Introduction

Chemotaxis, a kind of oriented motion of cells and organisms in response to
certain chemicals in the environment, 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 ([13, 24]). The
celebrated mathematical model describing chemotactic migration processes at
population level is Keller–Segel-type system of the form

$\left\\{\begin{array}[]{ll}u_{t}=\nabla\cdot(\gamma(u,v)\nabla
u-u\phi(u,v)\nabla v),&x\in\Omega,t>0,\\\ v_{t}=d\Delta
v-v+u,&x\in\Omega,t>0,\end{array}\right.$ (1.1)

in a bounded domain $\Omega\subset\mathbb{R}^{n}$ 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 ([17]). The most striking features of (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

$\left\\{\begin{array}[]{ll}u_{t}=\triangle u-\nabla\cdot(u\nabla
v),&x\in\Omega,t>0,\\\ v_{t}=d\Delta v-v+u,&x\in\Omega,t>0\end{array}\right.$
(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 ([9, 10, 34],
see also the surveys [2]). Apart from that, when $\phi$ and $\gamma$ in (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 [12, 32, 33]).

As a simplification of (1.1), Keller–Segel system with density dependent
motility

$\left\\{\begin{array}[]{ll}u_{t}=\nabla\cdot(\gamma(v)\nabla u-u\phi(v)\nabla
v),&x\in\Omega,t>0,\\\ v_{t}=d\Delta v-v+u,&x\in\Omega,t>0\end{array}\right.$
(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 [18]. Here the signal-dependent diffusivity
$\gamma(v)$ and chemotactic sensitivity function $\phi(v)$ are linked through

$\phi(v)=(\alpha-1)\gamma^{\prime}(v),$

where $\alpha\geq 0$ denotes the ratio of effective body length (i.e. distance
between the signal-receptors) to the walk length (see [19] for details).
Notice that when $\alpha=0$, there is only one receptor in a cell and hence
chemotaxis is driven by the undirect effect of chemicals in the absence of the
chemical gradient sensing. In this case, (1.3) reads as

$\left\\{\begin{array}[]{ll}u_{t}=\triangle(\gamma(v)u),&x\in\Omega,t>0,\\\
v_{t}=d\Delta v-v+u,&x\in\Omega,t>0,\end{array}\right.$ (1.4)

where the considered diffusion process of the population is essentially
Brownian, assumption $\gamma^{\prime}(v)<0$ accounts for the repressive effect
of the chemical concentration on the population motility ([5]). In the context
of acyl-homoserine lactone (AHL) density-dependent motility, the extended
model of (1.4)

$\left\\{\begin{array}[]{ll}u_{t}=\Delta(u\gamma(v))+\beta\displaystyle\frac{uw^{2}}{w^{2}+\lambda},&x\in\Omega,~{}t>0,\\\
v_{t}=D\Delta v+u-v,&x\in\Omega,~{}t>0,\\\ w_{t}=\Delta
w-\displaystyle\frac{uw^{2}}{w^{2}+\lambda},&x\in\Omega,~{}t>0\end{array}\right.$
(1.5)

was proposed in [20] to advocate that spatio-temporal pattern of Eeshcrichia
coli cells can be induced via so-called “self-trapping” mechanisms, that is at
low AHL level, 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 to the analysis caused by the degeneracy of
$\gamma(v)$ as $v\rightarrow\infty$ from the mathematical point of view.
Indeed, to the authors’s knowledge, Yoon and Kim ([38]) showed that in the
case of $\gamma(v)=\frac{c_{0}}{v^{k}}$ for small $c_{0}$, problem (1.4)
admits a global classical solutions in any dimensions. The smallness condition
on $c_{0}$ is removed lately in [1] for the parabolic-elliptic version of
(1.4) with $0<k<\frac{n}{(n-2)_{+}}$. Furthermore, for the full parabolic
system (1.4) in the three-dimensional setting, Tao and Winkler ([28]) showed
the existence of certain globally weak solutions, which become eventually
smooth and bounded for suitably small initial data $u_{0}$ under the
assumption

$\displaystyle(H)~{}~{}\gamma(v)\in C^{3}([0,\infty)),~{}\textnormal{and there
exist $\gamma_{1},~{}\gamma_{2},\eta>0$ such that
$0<\gamma_{1}\leq\gamma(v)\leq\gamma_{2},$}$ $|\gamma^{\prime}(v)|<\eta$ for
all $v\geq 0$.

It should be remarked that based on the comparison method, Fujie and Jiang
([8]) obtained the uniform-in-time boundedness to (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 analogue of (1.4) under the assumption $\displaystyle\max_{0\leq
v<+\infty}\frac{|\gamma^{\prime}(v)|^{2}}{\gamma(v)}<+\infty$ or
$\gamma(v)=v^{-k}$ with $0<k<\frac{n}{(n-2)_{+}}$ in [7, 14].

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 [24]. 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 ([5])

$\left\\{\begin{array}[]{ll}u_{t}=\triangle(\gamma(v)u)+\mu
u(1-u),&x\in\Omega,t>0,\\\ v_{t}=d\Delta
v-v+u,&x\in\Omega,t>0,\end{array}\right.$ (1.6)

it is shown in [15] that in two dimensional setting, the system admits a
unique global classical solution when the motility function $\gamma\in
C^{3}([0,\infty)),\gamma(v)>0$ and $\gamma^{\prime}(v)<0$ for all $v\geq 0$,
$\lim_{v\rightarrow\infty}\gamma(v)=0$ and
$\displaystyle\lim_{v\rightarrow\infty}\frac{\gamma^{\prime}(v)}{\gamma(v)}$
exists, and even the constant steady state $(1,1)$ is globally asymptotically
stable if $\mu>\frac{1}{16}\displaystyle\max_{0\leq
v<+\infty}\frac{|\gamma^{\prime}(v)|^{2}}{\gamma(v)}$. The global existence
thereof in the higher dimensions has been proved for large $\mu>0$ ([31]),
while for small $\mu$, the respective model can generate pattern formation
(see [23]). The reader is referred to [21, 22] for the other studies on the
related variants involving super-quadratic degradation terms.

In contexts of the diffusion of cells in a porous medium (see the discussions
in [3, 30]), Winkler ([37]) considered the cross–diffusion system

$\left\\{\begin{array}[]{ll}u_{t}=\triangle(\gamma(v)u^{m}),&x\in\Omega,t>0,\\\
v_{t}=\Delta v-v+u,&x\in\Omega,t>0\end{array}\right.$ (1.7)

in smoothly bounded convex domains $\Omega\subset\mathbb{R}^{n}$, where $m>1$,
$\gamma$ generalizes the prototype $\gamma(v)=a+b(v+d)^{-\alpha}$ with $a\geq
0,b>0,d\geq 0$ and $\alpha\geq 0$, 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
(1.7) are simultaneously enhanced.

In recent paper [16], Jin et al. considered the three-component system

$\left\\{\begin{array}[]{ll}u_{t}=\Delta(\gamma(v)u)+\beta uf(w)-\theta
u,&x\in\Omega,~{}t>0,\\\ v_{t}=D\Delta v+u-v,&x\in\Omega,~{}t>0,\\\
w_{t}=\Delta w-uf(w),&x\in\Omega,~{}t>0\end{array}\right.$ (1.8)

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

$f(w)\in
C^{1}([0,\infty)),~{}f(0)=0,~{}f(w)>0~{}\textnormal{in}~{}(0,\infty)~{}\textnormal{and}~{}f^{\prime}(w)>0~{}\textnormal{on}~{}[0,\infty).$
(1.9)

Based on the method of energy estimates and Moser iteration, they showed the
uniformly boundedness to initial–boundary value problem of (1.8), inter alia
the asymptotic behavior thereof when parameter $D$ is suitably large. 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 (1.8).
Specifically, we consider the asymptotic behavior to the initial–boundary
value problem

$\left\\{\begin{array}[]{ll}u_{t}=\Delta(\displaystyle\frac{u^{m}}{v^{\alpha}})+\beta
uf(w),&x\in\Omega,~{}t>0,\\\ v_{t}=D\Delta v+u-v,&x\in\Omega,~{}t>0,\\\
w_{t}=\Delta w-uf(w),&x\in\Omega,~{}t>0\end{array}\right.$ (1.10)

along with the initial conditions

$u(x,0)=u_{0},v(x,0)=v_{0}~{}~{}w(x,0)=w_{0},~{}~{}x\in\Omega$ (1.11)

and under the boundary conditions

$\displaystyle\frac{\partial u}{\partial\nu}=\displaystyle\frac{\partial
v}{\partial\nu}=\displaystyle\frac{\partial
w}{\partial\nu}=0~{}~{}\hbox{on}~{}\partial\Omega$ (1.12)

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

In what follows, for simplicity we shall drop the differential element in the
integrals without confusion, namely abbreviating $\int_{\Omega}f(x)dx$ as
$\int_{\Omega}f$ and $\int^{t}_{0}\int_{\Omega}f(x,\tau)dxd\tau$ as
$\int^{t}_{0}\int_{\Omega}f(\cdot,)d\tau$. With the assumption (1.9), our main
result asserts that the weak solutions approach the relevant homogeneous
steady state in the large time limit if $D$ is suitably large, which is stated
as follows.

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded convex domain with smooth
boundary, and suppose that $m>1,\alpha>0,\beta>0$ and $f$ satisfies (1.9).
Assume that initial data $(u_{0},v_{0},w_{0})\in(W^{1,\infty}(\Omega))^{3}$
with $u_{0}\gneqq 0,w_{0}\gneqq 0$ and $v_{0}>0$ in $\overline{\Omega}$. Then
problem (1.10)–(1.12) 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 $D>D_{0}$,

$\lim\limits_{t\rightarrow\infty}\|u(\cdot,t)-u_{\star}\|_{L^{\infty}(\Omega)}+\|v(\cdot,t)-u_{\star}\|_{L^{\infty}(\Omega)}+\|w(\cdot,t)\|_{L^{\infty}(\Omega)}=0$
(1.13)

with
$u_{\star}=\frac{1}{|\Omega|}\int_{\Omega}u_{0}+\frac{\beta}{|\Omega|}\int_{\Omega}w_{0}$.

As the first step to prove the above claim, in the next section it is shown
that problem (1.10)–(1.12) with $m>1$ and $\alpha>0$ possesses a globally
defined weak solution in two-dimensional setting by adjusting the argument in
[32]. With respect to the convergence properties asserted in (1.13), our
analysis is essentially different from that of [16]. In fact, thanks to
$\gamma_{1}\leq\gamma(v)\leq\gamma_{2}$ for all $v\geq 0$ in (H), authors of
[16] derived the estimate of $\|u(\cdot,t)\|_{L^{2}(\Omega)}$, which is the
start point of a priori estimate of $\|u(\cdot,t)\|_{L^{\infty}(\Omega)}$. In
particular, the assumption $\gamma_{1}\leq\gamma(v)$ plays an essential role
in constructing energy function
$\mathcal{F}(u,v):=\|u(\cdot,t)-u_{*}\|_{L^{2}(\Omega)}+\|v(\cdot,t)-u_{*}\|_{L^{2}(\Omega)}$,
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 [16] for the details). Our asymptotic
analysis consists at its core in an analysis of the functional

$\int_{\Omega}u^{2}+\eta\int_{\Omega}|\nabla v|^{2}$

for solutions of certain regularized versions of (1.10), 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
$\int^{\infty}_{3}\int_{\Omega}|\nabla u^{\frac{m+1}{2}}|^{2}$ and
$\int^{\infty}_{3}\int_{\Omega}|\nabla v|^{2}$ (see Lemma 5.3), and then
entails that as a consequence of these integral inequalities, all our
solutions asymptotically become homogeneous in space and hence satisfy (1.13)
(Lemmas 5.4–5.6).

######  Remark 1.1.

(1) Note that as an apparently inherent drawback, assumption (H) in [16]
excludes $\gamma(v)$ decay functions such as $v^{-\alpha}$. Indeed, despite
$v$ is bounded below by $\delta$ with the help of Lemma 2.3 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 existence of global solutions in [37], the
asymptotic behavior of solutions herein seems to be achieved for the higher-
dimensional version of (1.10) at the cost of additional constraint on $m$ and
$\alpha$.

##  2 Preliminaries

Throughout this paper, we shall pursue weak solutions to problem (1.10)–(1.12)
specified as follows.

######  Definition 2.1.

Let $m>1,\alpha>0,\beta>0$ and $f$ satisfies (1.9). Then a triple $(u,v,w)$ of
nonnegative functions

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

will be called a global weak solution of problem (1.10)–(1.12) if

$u^{m}/v^{\alpha}\in L^{1}_{loc}(\overline{\Omega}\times[0,\infty))$ (2.1)

and

$-\int^{\infty}_{0}\int_{\Omega}u\varphi_{t}-\int_{\Omega}u_{0}\varphi(\cdot,0)=\int^{\infty}_{0}\int_{\Omega}\frac{u^{m}}{v^{\alpha}}\Delta\varphi+\beta\int^{\infty}_{0}\int_{\Omega}uf(w)\varphi$
(2.2)

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

$-\int^{\infty}_{0}\int_{\Omega}v\varphi_{t}-\int_{\Omega}v_{0}\varphi(\cdot,0)=-D\int^{\infty}_{0}\int_{\Omega}\nabla
v\cdot\nabla\varphi-\int^{\infty}_{0}\int_{\Omega}v\varphi+\int^{\infty}_{0}\int_{\Omega}u\varphi$
(2.3)

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

$\int^{\infty}_{0}\int_{\Omega}w\varphi_{t}-\int_{\Omega}w_{0}\varphi(\cdot,0)=-\int^{\infty}_{0}\int_{\Omega}\nabla
w\cdot\nabla\varphi-\int^{\infty}_{0}\int_{\Omega}uf(w)\varphi$ (2.4)

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

Proceeding in a similar manner as done in [37], a global weak solution in the
above sense can be obtained as the limit of a sequence of solutions
$(u_{\varepsilon},v_{\varepsilon},w_{\varepsilon})$ of the regularized
problems

$\left\\{\begin{array}[]{lll}u_{\varepsilon
t}=\varepsilon\Delta(u_{\varepsilon}+1)^{M}+\Delta\left(u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}v_{\varepsilon}^{-\alpha}\right)+\beta
u_{\varepsilon}f(w_{\varepsilon}),&&x\in\Omega,~{}t>0,\\\ v_{\varepsilon
t}=D\Delta
v_{\varepsilon}+u_{\varepsilon}-v_{\varepsilon},&&x\in\Omega,~{}t>0,\\\
w_{\varepsilon t}=\Delta
w_{\varepsilon}-u_{\varepsilon}f(w_{\varepsilon}),&&x\in\Omega,~{}t>0,\\\
\frac{\partial u_{\varepsilon}}{\partial\nu}=\frac{\partial
v_{\varepsilon}}{\partial\nu}=\frac{\partial
w_{\varepsilon}}{\partial\nu}=0,&&x\in\partial\Omega,~{}t>0,\\\
u_{\varepsilon}(x,0)=u_{0},~{}v_{\varepsilon}(x,0)=v_{0},~{}w_{\varepsilon}(x,0)=w_{0},&&x\in\Omega\end{array}\right.$
(2.5)

with $M>m,\varepsilon\in(0,1)$.

######  Lemma 2.1.

Let $m>1,\alpha>0,\beta>0$ and $f$ satisfies (1.9). Then there exist
$(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ as well as nonnegative
functions

$\left\\{\begin{array}[]{lll}u\in
L^{\infty}(\overline{\Omega}\times[0,\infty))\\\ v\in
C^{0}(\overline{\Omega}\times[0,\infty))\bigcap
L^{2}_{loc}([0,\infty);W^{1,2}(\Omega))\\\ w\in
C^{0}(\overline{\Omega}\times[0,\infty))\bigcap
L^{2}_{loc}([0,\infty);W^{1,2}(\Omega))\end{array}\right.$ (2.6)

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

$\displaystyle u_{\varepsilon}\rightarrow u\quad\textnormal{\emph{a.e. in
$\Omega\times(0,\infty)$}},$ (2.7) $\displaystyle u_{\varepsilon}\rightarrow
u\quad\textnormal{\emph{in $\bigcap_{p\geq
1}L^{p}_{loc}(\overline{\Omega}\times[0,\infty))$ }},$ (2.8) $\displaystyle
v_{\varepsilon}\rightarrow v\quad\textnormal{\emph{in
$C_{loc}^{0}(\overline{\Omega}\times[0,\infty))$}},$ (2.9) $\displaystyle
w_{\varepsilon}\rightarrow w\quad\textnormal{\emph{in
$C_{loc}^{0}(\overline{\Omega}\times[0,\infty))$}},$ (2.10)
$\displaystyle\nabla v_{\varepsilon}\rightharpoonup\nabla
v\quad\textnormal{\emph{in
$L^{2}_{loc}(\overline{\Omega}\times[0,\infty))$}},$ (2.11)
$\displaystyle\nabla w_{\varepsilon}\rightharpoonup\nabla
w\quad\textnormal{\emph{in
$L^{2}_{loc}(\overline{\Omega}\times[0,\infty))$}}.$ (2.12)

Moreover, $v>0$ in $\overline{\Omega}\times(0,\infty)$ and $(u,v,w)$ forms a
global weak solution of (1.10)–(1.12) in the sense of Definition 2.1.

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 2.2.

Let $(u_{\varepsilon},v_{\varepsilon},w_{\varepsilon})$ be the classical
solution of (2.5) in $\Omega\times(0,\infty)$. Then we have

$\|u_{\varepsilon}(\cdot,t)\|_{L^{1}(\Omega)}+\beta\|w_{\varepsilon}(\cdot,t)\|_{L^{1}(\Omega)}=\|u_{0}\|_{L^{1}(\Omega)}+\beta\|w_{0}\|_{L^{1}(\Omega)},$
(2.13)
$\|u_{\varepsilon}(\cdot,t)\|_{L^{1}(\Omega)}\geq\|u_{0}\|_{L^{1}(\Omega)},$
(2.14)
$\int_{\Omega}v_{\varepsilon}(\cdot,t)\leq\int_{\Omega}v_{0}+\int_{\Omega}u_{0}+\beta\int_{\Omega}w_{0}$
(2.15)

as well as

$t\mapsto\|w_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}~{}\textnormal{\emph{is
nonincrasing in}}~{}~{}[0,\infty).$ (2.16)

###### Proof.

Multiplying $w_{\varepsilon}$-equation by $\beta$ and adding the result to
$u_{\varepsilon}$-equation in (2.5), we get

$\beta\frac{d}{dt}\int_{\Omega}w_{\varepsilon}+\frac{d}{dt}\int_{\Omega}u_{\varepsilon}=0,$
(2.17)

which immediately yields (2.13). An integration of the first equation in (2.5)
gives us

$\frac{d}{dt}\int_{\Omega}u_{\varepsilon}=\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})\geq
0$ (2.18)

which readily entails (2.14). Upon the integration of the second equation in
(2.5), we can see that

$\frac{d}{dt}\int_{\Omega}v_{\varepsilon}+\int_{\Omega}v_{\varepsilon}\leq\int_{\Omega}u_{\varepsilon}$

which, along with (2.13) leads to (2.15). Due to the fact that $f$ and
$w_{\varepsilon}$ are nonnegative, the claim in (2.16) results upon an
application of the maximum principle to $w_{\varepsilon}$-equation in (2.5). ∎

Let us first derive 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 ([11, 37]),
we present a proof of our results with some necessary details to make the
lower bound accessible to the sequel analysis.

######  Lemma 2.3.

If $D\geq 1$, then there exist a constant $\delta>0$ independent of $D$ such
that

$v_{\varepsilon}(x,t)>\delta$ (2.19)

for all $x\in\Omega$ and $t>2$.

###### Proof.

According to the pointwise lower bound estimate for the Neumann heat semigroup
$(e^{t\Delta})_{t\geq 0}$ on the convex domain $\Omega$, one can find
$c_{1}(\Omega)>0$ such that

$e^{t\Delta}\varphi\geq c_{1}(\Omega)\int_{\Omega}\varphi\qquad\textnormal{for
all $t\geq 1$ and each nonnegative $\varphi\in C^{0}(\overline{\Omega})$}$

(e.g. [6, 11]).

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

$\tilde{v}_{\tilde{t}}=\Delta\tilde{v}-D^{-1}\tilde{v}+D^{-1}u_{\varepsilon}(x,D^{-1}\tilde{t}).$
(2.20)

Now applying the variation-of-constant formula to (2.20), we have

$\begin{array}[]{ll}\tilde{v}(\cdot,\tilde{t})&=e^{\tilde{t}(\Delta-D^{-1})}v_{0}(\cdot)+D^{-1}\displaystyle\int^{\tilde{t}}_{0}e^{(\tilde{t}-s)(\Delta-D^{-1})}u_{\varepsilon}(\cdot,D^{-1}s)ds\\\
&\geq
D^{-1}\displaystyle\int^{\tilde{t}-1}_{0}e^{(\tilde{t}-s)(\Delta-D^{-1})}u_{\varepsilon}(\cdot,D^{-1}s)ds\\\
&\geq
c_{1}(\Omega)D^{-1}(\displaystyle\int^{\tilde{t}-1}_{0}e^{-D^{-1}(\tilde{t}-s)}ds)\inf\limits_{s\in(0,\infty)}\int_{\Omega}u_{\varepsilon}(\cdot,s)\\\
&\geq
c_{1}(\Omega)(e^{-D^{-1}}-e^{-D^{-1}\tilde{t}})\displaystyle\int_{\Omega}u_{0}\end{array}$
(2.21)

for all $\tilde{t}>2$. Hence due to $D\geq 1$, we can see that for
$x\in\Omega$ and $\tilde{t}\geq 2D$

$\tilde{v}(x,\tilde{t})\geq\frac{c_{1}(\Omega)}{2e}\int_{\Omega}u_{0},$

and readily establish (2.19) with
$\delta=\frac{c_{1}(\Omega)}{2e}\int_{\Omega}u_{0}$. ∎

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

######  Lemma 2.4.

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

$\|v_{\varepsilon}(\cdot,t)\|_{L^{p}(\Omega)}\leq C(p)(1+D^{\frac{1}{p}-1})$
(2.22)

for all $t>2$.

###### Proof.

Applying a Duhamel’s formula to (2.20) and employing well-known smoothing
properties of the Neumann heat semigroup $(e^{t\Delta})_{t\geq 0}$ on $\Omega$
(see Lemma 3 of [26] or Lemma 1.3 of [35] for example), we can find $c_{p}>0$
such that for any $\tilde{t}\geq 2D$

$\begin{array}[]{rl}&\|\tilde{v}_{\varepsilon}(\cdot,\tilde{t})\|_{L^{p}(\Omega)}\\\
=&\|e^{-D^{-1}\tilde{t}}e^{\tilde{t}\Delta}v_{0}(\cdot)+D^{-1}\displaystyle\int^{\tilde{t}}_{0}e^{(\tilde{t}-s)(\Delta-D^{-1})}u_{\varepsilon}(\cdot,D^{-1}s)ds\|_{L^{p}(\Omega)}\\\
\leq&c_{p}e^{-D^{-1}\tilde{t}}(1+\tilde{t}^{-1+\frac{1}{p}})\|v_{0}\|_{L^{1}(\Omega)}+\displaystyle\frac{c_{p}}{D}\displaystyle\int^{\tilde{t}}_{0}e^{-D^{-1}(\tilde{t}-s)}(1+(\tilde{t}-s)^{-1+\frac{1}{p}})\|u_{\varepsilon}(\cdot,D^{-1}s)\|_{L^{1}(\Omega)}ds\\\
\leq&2c_{p}e^{-D^{-1}\tilde{t}}\|v_{0}\|_{L^{1}(\Omega)}+\displaystyle\frac{c_{p}}{D}(\|u_{0}\|_{L^{1}(\Omega)}+\beta\|w_{0}\|_{L^{1}(\Omega)})\displaystyle\int^{\tilde{t}}_{0}e^{-D^{-1}(\tilde{t}-s)}(1+(\tilde{t}-s)^{-1+\frac{1}{p}})ds\\\
=&2c_{p}e^{-D^{-1}\tilde{t}}\|v_{0}\|_{L^{1}(\Omega)}+\displaystyle\frac{c_{p}}{D}(\|u_{0}\|_{L^{1}(\Omega)}+\beta\|w_{0}\|_{L^{1}(\Omega)})\displaystyle\int^{\tilde{t}}_{0}e^{-D^{-1}\sigma}(1+\sigma^{-1+\frac{1}{p}})d\sigma\\\
\leq&2c_{p}e^{-D^{-1}\tilde{t}}\|v_{0}\|_{L^{1}(\Omega)}+\displaystyle\frac{c_{p}}{D}(\|u_{0}\|_{L^{1}(\Omega)}+\beta\|w_{0}\|_{L^{1}(\Omega)})(D+D^{\frac{1}{p}}\displaystyle\int^{\infty}_{0}e^{-\sigma}\sigma^{-1+\frac{1}{p}}d\sigma)\\\
\leq&2c_{p}\|v_{0}\|_{L^{1}(\Omega)}+(1+D^{\frac{1}{p}-1})c_{p}(\|u_{0}\|_{L^{1}(\Omega)}+\beta\|w_{0}\|_{L^{1}(\Omega)})(1+\displaystyle\int^{\infty}_{0}e^{-\sigma}\sigma^{-1+\frac{1}{p}}d\sigma)\end{array}$

which ends up (2.22) with
$C(p)=3c_{p}(\|u_{0}\|_{L^{1}(\Omega)}+\beta\|w_{0}\|_{L^{1}(\Omega)})(1+\displaystyle\int^{\infty}_{0}e^{-\sigma}\sigma^{-1+\frac{1}{p}}d\sigma)$.
∎

##  3 Space-time $L^{1}$-estimates for
$u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon}$

In this section, taking advantage of special structure of the diffusive
processes in (2.5) (also (1.8)), the classical duality arguments (cf. [28, 4])
is used to obtain the fundamental regularity information for a bootstrap
argument. To this end, we denote $A$ to the self-adjoint realization of
$-\Delta+1$ under homogeneous Neumann boundary condition in $L^{2}(\Omega)$
with its domain given by $D(A)=\left\\{\varphi\in
W^{2,2}(\Omega)|\frac{\partial\varphi}{\partial\nu}=0\right\\}$ and $A$ is
self-adjoint and possesses a family $(A^{\beta})_{\beta\in\mathbb{R}}$ of
corresponding densely defined self-adjoint fractional powers.

######  Lemma 3.1.

Assume that $m>1$ and $D\geq 1$, then for all $t>2$

$\frac{d}{dt}\int_{\Omega}|A^{-\frac{1}{2}}(u_{\varepsilon}+1)|^{2}+\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon}\leq
C\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{m+1}+C$ (3.1)

with constant $C>0$ independent of $D$.

###### Proof.

Due to $\partial_{t}(u_{\varepsilon}+1)=u_{\varepsilon t}$, the first equation
in (2.5) can be written as

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}A^{-1}(u_{\varepsilon}+1)+\varepsilon(u_{\varepsilon}+1)^{M}+u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}v_{\varepsilon}^{-\alpha}\\\
=&A^{-1}\left\\{\varepsilon(u_{\varepsilon}+1)^{M}+u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}v_{\varepsilon}^{-\alpha}+\beta
u_{\varepsilon}f(w_{\varepsilon})\right\\}.\end{array}$ (3.2)

Testing (3.2) by $u_{\varepsilon}+1$, one has

$\begin{array}[]{rl}&\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|A^{-\frac{1}{2}}(u_{\varepsilon}+1)|^{2}+\varepsilon\int_{\Omega}(u_{\varepsilon}+1)^{M+1}+\int_{\Omega}u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}(u_{\varepsilon}+1)v_{\varepsilon}^{-\alpha}\\\
=&\varepsilon\displaystyle\int_{\Omega}(u_{\varepsilon}+1)^{M}A^{-1}(u_{\varepsilon}+1)+\int_{\Omega}u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}v_{\varepsilon}^{-\alpha}A^{-1}(u_{\varepsilon}+1)+\beta\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})A^{-1}(u_{\varepsilon}+1).\end{array}$
(3.3)

Thanks to $W^{2,2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ in two-
dimensional setting and the standard elliptic regularity in $L^{2}(\Omega)$,
one can find $c_{1}>0$ and $c_{2}>0$ such that

$\|\varphi\|^{M+1}_{L^{M+1}(\Omega)}\leq
c_{1}\|\varphi\|^{M+1}_{W^{2,2}(\Omega)}\leq
c_{2}\|A\varphi\|^{M+1}_{L^{2}(\Omega)}$ (3.4)

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

$\begin{array}[]{rl}\varepsilon\displaystyle\int_{\Omega}(u_{\varepsilon}+1)^{M}A^{-1}(u_{\varepsilon}+1)\leq&\displaystyle\frac{\varepsilon}{2}\int_{\Omega}(u_{\varepsilon}+1)^{M+1}+\displaystyle\frac{\varepsilon}{2}\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{M+1}\\\
\leq&\displaystyle\frac{\varepsilon}{2}\|u_{\varepsilon}+1\|^{M+1}_{L^{M+1}(\Omega)}+\displaystyle\frac{\varepsilon
c_{1}}{2}\|A^{-1}(u_{\varepsilon}+1)\|^{M+1}_{W^{2,2}(\Omega)}\\\
=&\displaystyle\frac{\varepsilon}{2}\displaystyle\int_{\Omega}(u_{\varepsilon}+1)^{M+1}+\displaystyle\frac{\varepsilon
c_{1}c_{2}}{2}\|u_{\varepsilon}+1\|^{M+1}_{L^{2}(\Omega)}\end{array}$ (3.5)

which, along with the interpolation inequality that for any
$\varepsilon_{1}>0$, there exits $c(\varepsilon_{1})>0$ such that
$\|\varphi\|_{L^{2}(\Omega)}\leq\varepsilon_{1}\|\varphi\|_{L^{M+1}(\Omega)}+c(\varepsilon_{1})\|\varphi\|_{L^{1}(\Omega)}$
due to $M>1$, entails that

$\varepsilon\displaystyle\int_{\Omega}(u_{\varepsilon}+1)^{M}A^{-1}(u_{\varepsilon}+1)\leq\displaystyle\frac{3\varepsilon}{4}\displaystyle\int_{\Omega}(u_{\varepsilon}+1)^{M+1}+c_{3}\|u_{\varepsilon}+1\|^{M+1}_{L^{1}(\Omega)}$
(3.6)

Furthermore, since
$\|w_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\|w_{0}\|_{L^{\infty}(\Omega)}$,
we use Lemma 2.1 and Young’s inequality to get that for $t>2$,

$\begin{array}[]{rl}&\displaystyle\int_{\Omega}u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}v_{\varepsilon}^{-\alpha}A^{-1}(u_{\varepsilon}+1)\\\\[5.69054pt]
\leq&\displaystyle\frac{1}{4}\displaystyle\int_{\Omega}\left\\{u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}\right\\}^{\frac{m+1}{m}}v_{\varepsilon}^{-\alpha}+c_{4}\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{m+1}v_{\varepsilon}^{-\alpha}\\\\[5.69054pt]
\leq&\displaystyle\frac{1}{4}\int_{\Omega}u^{\frac{m+1}{m}}_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{\frac{m^{2}-1}{m}}v_{\varepsilon}^{-\alpha}+c_{4}\delta^{-\alpha}\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{m+1}\end{array}$
(3.7)

and

$\begin{array}[]{rl}&\beta\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})A^{-1}(u_{\varepsilon}+1)\\\\[8.53581pt]
\leq&\displaystyle\frac{1}{4}\int_{\Omega}u_{\varepsilon}^{m+1}v_{\varepsilon}^{-\alpha}+c_{5}\int_{\Omega}v_{\varepsilon}^{\frac{\alpha}{m}}|A^{-1}(u_{\varepsilon}+1)|^{\frac{m+1}{m}}\\\
\leq&\displaystyle\frac{1}{4}\int_{\Omega}u^{m+1}_{\varepsilon}v_{\varepsilon}^{-\alpha}+\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{m+1}+c_{6}\int_{\Omega}v_{\varepsilon}^{\frac{\alpha}{m-1}}.\end{array}$
(3.8)

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

$\int_{\Omega}u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}(u_{\varepsilon}+1)v_{\varepsilon}^{-\alpha}\geq\frac{1}{4}\int_{\Omega}u_{\varepsilon}^{\frac{m+1}{m}}(u_{\varepsilon}+\varepsilon)^{\frac{m^{2}-1}{m}}v_{\varepsilon}^{-\alpha}+\frac{3}{4}\int_{\Omega}u_{\varepsilon}^{m+1}v_{\varepsilon}^{-\alpha},$

and hence insert (3.8) and (LABEL:3.7) into (3.3) to get

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\int_{\Omega}|A^{-\frac{1}{2}}(u_{\varepsilon}+1)|^{2}+\int_{\Omega}u_{\varepsilon}^{m+1}v_{\varepsilon}^{-\alpha}\\\
\leq&2(c_{4}\delta^{-\alpha}+1)\displaystyle\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{m+1}+2c_{6}\int_{\Omega}v_{\varepsilon}^{\frac{\alpha}{m-1}},\end{array}$

which along with Lemma 2.4 and $D\geq 1$ readily arrive at (3.1). ∎

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

$\int^{t+1}_{t}\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon}\leq
C$

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

######  Lemma 3.2.

If $m>1$ and $D\geq 1$, then there exists $C>0$ independent of $D$ such that

$\int^{t+1}_{t}\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon}\leq
C~{}~{}~{}~{}~{}\hbox{for all}~{}~{}t>2.$ (3.9)

###### Proof.

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

$\displaystyle\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{m+1}\leq
c_{1}\displaystyle\|u_{\varepsilon}+1\|_{L^{2}(\Omega)}^{m+1}.$

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

$c_{1}\|u_{\varepsilon}+1\|^{m+1}_{L^{2}(\Omega)}\leq\eta\|u_{\varepsilon}+1\|^{m+1}_{L^{p}(\Omega)}+c_{1}(\eta)\|u_{\varepsilon}+1\|^{m+1}_{L^{1}(\Omega)}.$

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

$\displaystyle\int_{\Omega}u^{p}_{\varepsilon}$ $\displaystyle=$
$\displaystyle\int_{\Omega}\left(u^{m+1}_{\varepsilon}v_{\varepsilon}^{-\alpha}\right)^{\frac{p}{m+1}}v_{\varepsilon}^{\frac{p\alpha}{m+1}}$
$\displaystyle\leq$
$\displaystyle(\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon})^{\frac{p}{m+1}}(\int_{\Omega}v^{\frac{p\alpha}{m+1-p}}_{\varepsilon})^{\frac{m+1-p}{m+1}}.$

Hence combining above estimates with Lemma 2.4, we arrive at

$\begin{array}[]{rl}\displaystyle\int_{\Omega}|A^{-1}(u_{\varepsilon}+1)|^{m+1}&\leq\eta\displaystyle\|u_{\varepsilon}+1\|_{L^{p}(\Omega)}^{m+1}+c_{1}(\eta)\|u_{\varepsilon}+1\|^{m+1}_{L^{1}(\Omega)}\\\
&\leq\eta\displaystyle\|u_{\varepsilon}\|_{L^{p}(\Omega)}^{m+1}+c_{2}(\eta)\\\
&\leq\displaystyle\eta(\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon})(\int_{\Omega}v^{\frac{p\alpha}{m+1-p}}_{\varepsilon})^{\frac{m+1-p}{p}}+c_{2}(\eta)\\\
&\leq\displaystyle\eta
c_{3}(\alpha,m)(\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon})+c_{2}(\eta)~{}~{}~{}\hbox{for
all}~{}~{}t>2.\end{array}$ (3.10)

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

$\begin{array}[]{rl}\displaystyle\int_{\Omega}|A^{-\frac{1}{2}}(u_{\varepsilon}+1)|^{2}=&\displaystyle\int_{\Omega}(u_{\varepsilon}+1)A^{-1}(u_{\varepsilon}+1)\\\
\leq&\displaystyle\|u_{\varepsilon}+1\|_{L^{2}(\Omega)}\|A^{-1}(u_{\varepsilon}+1)\|_{L^{2}(\Omega)}\\\
\leq&c_{4}\displaystyle\|u_{\varepsilon}+1\|^{2}_{L^{2}(\Omega)}\\\
\leq&c_{5}\displaystyle\|u_{\varepsilon}\|^{2}_{L^{2}(\Omega)}+c_{5}\\\
\leq&\displaystyle
c_{6}\|u_{\varepsilon}\|^{m+1}_{L^{2}(\Omega)}+c_{6}.\end{array}$ (3.11)

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

$\displaystyle\int_{\Omega}|A^{-\frac{1}{2}}(u_{\varepsilon}+1)|^{2}\leq\displaystyle\eta
c_{3}(\alpha,m)(\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon})+c_{7}(\eta).$
(3.12)

Therefore inserting (3.10) and (3.12) into (3.1) and taking $\eta$
sufficiently small, we have

$\frac{d}{dt}\int_{\Omega}|A^{-\frac{1}{2}}(u_{\varepsilon}+1)|^{2}+c_{8}\int_{\Omega}|A^{-\frac{1}{2}}(u_{\varepsilon}+1)|^{2}+c_{8}\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon}\leq
c_{9}$ (3.13)

for some $c_{8}>0,c_{9}>0$ which are independent of $D$. Furthermore, by Lemma
3.4 of [27] , we immediately obtain (3.9). ∎

As the direct consequence of Lemma 3.2 and Lemma 2.4, we have

######  Lemma 3.3.

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

$\int^{t+1}_{t}\int_{\Omega}u^{p}_{\varepsilon}(\cdot,s)ds\leq
C(p)~{}~{}~{}\hbox{for all}~{}~{}t>2.$ (3.14)

###### Proof.

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

$\begin{array}[]{rl}\displaystyle\int^{t+1}_{t}\int_{\Omega}u^{p}_{\varepsilon}&=\displaystyle\int^{t+1}_{t}\int_{\Omega}\left(u^{m+1}_{\varepsilon}v_{\varepsilon}^{-\alpha}\right)^{\frac{p}{m+1}}v_{\varepsilon}^{\frac{p\alpha}{m+1}}\\\
&\leq\displaystyle\int^{t+1}_{t}\int_{\Omega}u^{m+1}_{\varepsilon}v^{-\alpha}_{\varepsilon}+\int^{t+1}_{t}\int_{\Omega}v^{\frac{p\alpha}{m+1-p}}_{\varepsilon},\end{array}$

which leads to (3.14) with the help of Lemma 2.4. ∎

##  4 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 4.1.

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

$\|v_{\varepsilon}(\cdot,t)\|_{W^{1,q}(\Omega)}\leq C$ (4.1)

as well as

$\|w_{\varepsilon}(\cdot,t)\|_{W^{1,q}(\Omega)}\leq C$ (4.2)

for all $t>2$.

###### Proof.

From the continuity of function $h(x)=\frac{2x}{(4-x)_{+}}$, 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

$\frac{p}{p-1}\cdot(\frac{1}{2}+\frac{1}{p}-\frac{1}{q})<1.$ (4.3)

From the smoothing properties of Neumann heat semigroup $(e^{t\Delta})_{t\geq
0}$, it follow that there exist $c_{i}>0(i=1,2)$ such that

$\|e^{\Delta}\varphi\|_{W^{1,q}(\Omega)}\leq
c_{1}\|\varphi\|_{L^{1}(\Omega)}\quad\textnormal{\emph{for all $t\in(0,1)$ and
$\varphi\in C^{0}(\overline{\Omega})$}}$ (4.4)

as well as

$\|e^{t\Delta}\varphi\|_{W^{1,q}(\Omega)}\leq
c_{2}t^{-\frac{1}{2}-\frac{1}{2}(\frac{1}{p}-\frac{1}{q})}\|\varphi\|_{L^{p}(\Omega)}\quad\textnormal{\emph{for
all $t\in(0,1)$ and $\varphi\in C^{0}(\overline{\Omega})$}}.$ (4.5)

Therefore by the Duhamel representation to the second equation of (2.20), we
obtain

$\displaystyle\|\tilde{v}_{\varepsilon}(\cdot,t)\|_{W^{1,q}(\Omega)}$
$\displaystyle=\left\|e^{(\Delta-D^{-1})}\tilde{v}_{\varepsilon}(\cdot,t-1)+D^{-1}\displaystyle\int^{t}_{t-1}e^{(t-s)(\Delta-D^{-1})}u_{\varepsilon}(\cdot,D^{-1}s)ds\right\|_{W^{1,q}(\Omega)}$
(4.6)
$\displaystyle\leq\|e^{\Delta}\tilde{v}_{\varepsilon}(\cdot,t-1)\|_{W^{1,q}(\Omega)}+D^{-1}\displaystyle\int^{t}_{t-1}\|e^{(t-s)\Delta}u_{\varepsilon}(\cdot,D^{-1}s)\|_{W^{1,q}(\Omega)}ds.$

Due to (4.4) and (4.5), we have

$\|e^{\Delta}\tilde{v}_{\varepsilon}(\cdot,t-1)\|_{W^{1,q}(\Omega)}\leq
c_{1}\|\tilde{v}_{\varepsilon}(\cdot,t-1)\|_{L^{1}(\Omega)}$ (4.7)

and

$\begin{array}[]{rl}&\displaystyle\int^{t}_{t-1}\|e^{-(t-s)\Delta}u_{\varepsilon}(\cdot,D^{-1}s)\|_{W^{1,q}(\Omega)}\\\
\leq&c_{2}\displaystyle\int^{t}_{t-1}(t-s)^{-\frac{1}{2}-(\frac{1}{p}-\frac{1}{q})}\|u_{\varepsilon}(\cdot,D^{-1}s)\|_{L^{p}(\Omega)}ds\\\
\leq&c_{2}\displaystyle\left\\{\int^{t}_{t-1}(t-s)^{{}^{-\frac{p}{p-1}(\frac{1}{2}+\frac{1}{p}-\frac{1}{q})}}ds\right\\}^{\frac{p-1}{p}}\left\\{\int^{t}_{t-1}\|u_{\varepsilon}(\cdot,D^{-1}s)\|_{L^{p}(\Omega)}^{p}ds\right\\}^{\frac{1}{p}}\\\
\leq&c_{2}\displaystyle(\int^{1}_{0}\sigma^{{}^{-\frac{p}{p-1}\cdot\left(\frac{1}{2}+\frac{1}{p}-\frac{1}{q}\right)}}d\sigma)^{\frac{p-1}{p}}\left\\{\int^{t}_{t-1}\|u_{\varepsilon}(\cdot,D^{-1}s)\|_{L^{p}(\Omega)}^{p}ds\right\\}^{\frac{1}{p}}\\\
\leq&c_{2}D^{\frac{1}{p}}\displaystyle(\int^{1}_{0}\sigma^{{}^{-\frac{p}{p-1}\cdot\left(\frac{1}{2}+\frac{1}{p}-\frac{1}{q}\right)}}d\sigma)^{\frac{p-1}{p}}\left\\{\int^{D^{-1}t}_{D^{-1}t-D^{-1}}\|u_{\varepsilon}(\cdot,s)\|_{L^{p}(\Omega)}^{p}ds\right\\}^{\frac{1}{p}}\\\
\leq&c_{3}D^{\frac{1}{p}},\end{array}$ (4.8)

where due to $D\geq 1$ and the application of Lemma 3.3, we have

$\int^{D^{-1}t}_{D^{-1}t-D^{-1}}\|u_{\varepsilon}(\cdot,s)\|_{L^{p}(\Omega)}^{p}ds\leq\int^{D^{-1}t}_{D^{-1}t-1}\|u_{\varepsilon}(\cdot,s)\|_{L^{p}(\Omega)}^{p}ds\leq
c_{4}$

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 (4.3). Hence combining (4.6) with (4.7) and (4.8) gives

$\begin{array}[]{rl}\|v_{\varepsilon}(\cdot,t)\|_{W^{1,q}(\Omega)}&\leq
c_{2}\|\tilde{v}_{\varepsilon}(\cdot,t-1)\|_{L^{1}(\Omega)}+c_{3}D^{\frac{1}{p}-1}\\\
&\leq\displaystyle
c_{2}(\int_{\Omega}u_{0}+\beta\int_{\Omega}w_{0})+c_{3}\end{array}$

for all $t>2$ and thus completes the proof of (4.1).

Next due to
$\|w_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\|w_{0}\|_{L^{\infty}(\Omega)}$,
an application of Duhamel representation to the third equation in (2.5) yields

$\|w_{\varepsilon}(\cdot,t)\|_{W^{1,q}(\Omega)}\leq\left\|e^{\Delta}w_{\varepsilon}\left(\cdot,t-1\right)\right\|_{W^{1,q}(\Omega)}\\\
+f(\|w_{0}\|_{L^{\infty}(\Omega)})\int^{t}_{t-1}\left\|e^{(t-s)\Delta}u_{\varepsilon}(\cdot,s)\right\|_{W^{1,q}(\Omega)}ds,$

and thereby (4.2) 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 4.2.

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

$\begin{array}[]{rl}&\displaystyle\frac{1}{p}\displaystyle\int_{\Omega}\frac{d}{dt}(u_{\varepsilon}+\varepsilon)^{p}\cdot\varphi+(p-1)M\varepsilon\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p-2}(u_{\varepsilon}+1)^{M-1}|\nabla
u_{\varepsilon}|^{2}\varphi\\\
=&(1-p)\displaystyle\int_{\Omega}(mu_{\varepsilon}+\varepsilon)(u_{\varepsilon}+\varepsilon)^{m+p-4}|\nabla
u_{\varepsilon}|^{2}\varphi+\alpha(p-1)\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+p-3}v_{\varepsilon}^{-\alpha-1}\nabla
u_{\varepsilon}\cdot\nabla v_{\varepsilon}\varphi\\\
&+(1-p)\displaystyle\int_{\Omega}(mu_{\varepsilon}+\varepsilon)(u_{\varepsilon}+\varepsilon)^{m+p-3}v_{\varepsilon}^{-\alpha}\nabla
u_{\varepsilon}\cdot\nabla\varphi-M\varepsilon\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p-1}(u_{\varepsilon}+1)^{M-1}\nabla
u_{\varepsilon}\cdot\nabla\varphi\\\
&+\alpha\displaystyle\int_{\Omega}u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m+p-2}v_{\varepsilon}^{-\alpha-1}\nabla
v_{\varepsilon}\cdot\nabla\varphi+\beta\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})(u_{\varepsilon}+\varepsilon)^{p-1}\varphi\end{array}$
(4.9)

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

###### Proof.

This can be verified by straightforward computation. ∎

Thanks to the boundedness of $\|\nabla
v_{\varepsilon}(\cdot,t)\|_{L^{q}(\Omega)}$ with some $q>2$ in Lemma 4.1, we
can achieve the following $D$-independent $L^{p}$-estimate of
$u_{\varepsilon}$ with finite $p$.

######  Lemma 4.3.

Let $m>1$ and $D\geq 1$. Then for any $p>1$, there exist $t_{0}\geq 2$ and
constant $C(p)>0$ such that

$\|u_{\varepsilon}(\cdot,t)\|_{L^{p}(\Omega)}\leq C(p)$ (4.10)

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

###### Proof.

According to Lemma 2.3 and Lemma 2.4, one can find $c_{i}>0(i=1,2)$
independent of $D\geq 1$ fulfilling

$v_{\varepsilon}^{-\alpha}(x,t)\geq c_{1},\quad
v_{\varepsilon}^{-\alpha-2}(x,t)\leq
c_{2}~{}~{}\hbox{in}~{}~{}\Omega\times(2,\infty)$ (4.11)

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

Letting $\varphi\equiv 1$ in (4.9) and by Young’s inequality, we have

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}+p(p-1)\displaystyle\int_{\Omega}(mu_{\varepsilon}+\varepsilon)(u_{\varepsilon}+\varepsilon)^{m+p-4}v_{\varepsilon}^{-\alpha}|\nabla
u_{\varepsilon}|^{2}+\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}\\\
\leq&\alpha
p(p-1)\displaystyle\int_{\Omega}u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m+p-3}v_{\varepsilon}^{-\alpha-1}\nabla
u_{\varepsilon}\cdot\nabla v_{\varepsilon}+\beta
p\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})(u_{\varepsilon}+\varepsilon)^{p-1}+\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}\\\
\leq&\displaystyle\frac{p(p-1)}{2}\displaystyle\int_{\Omega}(mu_{\varepsilon}+\varepsilon)(u_{\varepsilon}+\varepsilon)^{m+p-4}v_{\varepsilon}^{-\alpha}|\nabla
u_{\varepsilon}|^{2}+\displaystyle\frac{\alpha^{2}p(p-1)}{2}\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+p-1}v_{\varepsilon}^{-\alpha-2}|\nabla
v_{\varepsilon}|^{2}+\\\ &+\beta
pf(\|w_{0}\|_{L^{\infty}(\Omega)})\displaystyle\int_{\Omega}u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{p-1}+\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}\end{array}$

Furthermore, recalling (4.11), we can find $c_{3}>0$ and $c_{4}>0$ independent
of $p$ such that

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}+c_{3}\displaystyle\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+p-1}{2}}|^{2}+\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}\\\
\leq&c_{4}p^{2}\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+p-1}|\nabla
v_{\varepsilon}|^{2}+c_{4}p\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}.\end{array}$
(4.12)

According to (4.1), $\|\nabla v_{\varepsilon}\|^{2}_{L^{q}(\Omega)}\leq c_{5}$
for any fix $q\in(2,\frac{2(m+1)}{(3-m)_{+}})$, and hence the Hölder
inequality yields

$\begin{array}[]{rl}&c_{4}p^{2}\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+p-1}|\nabla
v_{\varepsilon}|^{2}\\\
\leq&c_{4}p^{2}\left\\{\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{\frac{(m+p-1)q}{q-2}}\right\\}^{1-\frac{2}{q}}\|\nabla
v_{\varepsilon}\|^{2}_{L^{q}(\Omega)}\\\
\leq&c_{4}c_{5}p^{2}\|(u_{\varepsilon}+\varepsilon)^{\frac{m+p-1}{2}}\|^{2}_{L^{\frac{2q}{q-2}}(\Omega)}\\\
\leq&\displaystyle\frac{c_{3}}{4}\displaystyle\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+p-1}{2}}|^{2}+c_{6}(p),\end{array}$
(4.13)

where we have used an Ehrling-type inequality due to
$W^{1,2}(\Omega)\hookrightarrow L^{\frac{2q}{q-2}}(\Omega)$ in two-dimensional
setting and (2.13).

On the other hand, since

$c_{4}p\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}\leq\eta\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+p-1}+c(\eta)=\eta\|(u_{\varepsilon}+\varepsilon)^{\frac{m+p-1}{2}}\|^{2}_{L^{2}(\Omega)}+c(\eta)$

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

$c_{4}p\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}\leq\displaystyle\frac{c_{3}}{4}\displaystyle\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+p-1}{2}}|^{2}+c_{7}(p).$
(4.14)

Now inserting (4.14) and (4.13) into (4.12), we infer that for all $t\geq 2$

$\frac{d}{dt}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}+\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{p}\leq
c_{8}(p)$ (4.15)

with $c_{8}(p)>0$ independent of $D\geq 1$, which along with a standard
comparison argument implies that there exists $t_{0}>2$ such that

$\int_{\Omega}(u_{\varepsilon}(\cdot,t)+\varepsilon)^{p}\leq 2c_{8}(p)$ (4.16)

for all $t\geq t_{0}$ 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
[29] for example) and we list the result here without proof.

######  Lemma 4.4.

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

$\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C$ (4.17)

for all $t>t_{0}$ with $t_{0}$ as in Lemma 4.3.

######  Remark 4.1.

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 [37] for reference). However, the explicit dependence
of $\|u_{\varepsilon}(\cdot,t)\|_{L^{p}(\Omega)}$ on $D$ is required to
investigate the large time behavior of solutions in the sequel. Hence $D\geq
1$ 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 4.1.

######  Lemma 4.5.

Let $m>1,\alpha>0$ and suppose that $D\geq 1$. Then there exists constant
$C>0$ independent of $D$ and $\varepsilon$ such that

$\|\nabla v_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C$ (4.18)

as well as

$\|\nabla w_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C$ (4.19)

for all $t>t_{0}+1$.

###### Proof.

Due to $\|\nabla
e^{\Delta}\tilde{v}(\cdot,\tilde{t}-1)\|_{L^{\infty}(\Omega)}\leq
c_{1}\|\tilde{v}(\cdot,\tilde{t}-1)\|_{L^{1}(\Omega)}$, as the proof of Lemma
4.1, we use Duhamel formula of (2.20) in the following way

$\begin{array}[]{rl}\|\nabla\tilde{v}(\cdot,\tilde{t})\|_{L^{\infty}(\Omega)}&=\left\|\nabla
e^{(\Delta-D^{-1})}\tilde{v}(\cdot,\tilde{t}-1)+D^{-1}\displaystyle\int^{\tilde{t}}_{\tilde{t}-1}\nabla
e^{(t-s)(\Delta-D^{-1})}u(\cdot,D^{-1}s)ds\right\|_{L^{\infty}(\Omega)}\\\
&\leq\|\nabla
e^{\Delta}\tilde{v}(\cdot,\tilde{t}-1)\|_{L^{\infty}(\Omega)}+D^{-1}\displaystyle\int^{\tilde{t}}_{\tilde{t}-1}\|\nabla
e^{(\tilde{t}-s)\Delta}u(\cdot,D^{-1}s)\|_{L^{\infty}(\Omega)}ds\\\ &\leq
c_{1}\|\tilde{v}(\cdot,\tilde{t}-1)\|_{L^{1}(\Omega)}+c_{2}D^{-1}\displaystyle\int^{\tilde{t}}_{\tilde{t}-1}(1+(\tilde{t}-s)^{-\frac{3}{4}})ds\max_{\tilde{t}-1\leq
s\leq\tilde{t}}\|u(\cdot,D^{-1}s)\|_{L^{4}(\Omega)}.\end{array}$

for all $\tilde{t}>t_{0}D+1$, which along with (4.17) readily leads to (4.18).
It is obvious that (4.19) can be proved similarly. ∎

##  5 Asymptotic behavior

###  5.1 Weak decay information

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

######  Lemma 5.1.

Let $(u,v,w)$ be the nonnegative global solution of (1.10)–(1.12) obtained in
Lemma 2.1. Then there exist $\kappa\in(0,1)$ and $C>0$ such that for all
$t>t_{0}$

$\|u\|_{C^{\kappa,\frac{\kappa}{2}}(\overline{\Omega}\times[t,t+1])}\leq C$
(5.1)

as well as

$\|v\|_{C^{2+\kappa,1+\frac{\kappa}{2}}(\overline{\Omega}\times[t,t+1])}+\|w\|_{C^{2+\kappa,1+\frac{\kappa}{2}}(\overline{\Omega}\times[t,t+1])}\leq
C.$ (5.2)

###### Proof.

We rewrite the first equation of (2.5) 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 2.3, 2.4, 4.4 and 4.5, there exist two constants $c_{1}>0$
and $c_{2}>0$ independent of $D\geq 1$ satisfying

$c_{1}\leq v_{\varepsilon}^{-\alpha}(x,t)\leq
c_{2}~{}~{}\hbox{in}~{}~{}\Omega\times(t_{0},\infty)$ (5.3)

and

$\|v_{\varepsilon}^{-\alpha-1}(\cdot,t)\|_{L^{\infty}(\Omega)}+\|\nabla
v_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+\|w_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
c_{2}~{}~{}\hbox{for}~{}~{}t\geq t_{0}.$

This guarantees that for all $(x,t)\in\Omega\times(t_{0},\infty)$

$a(x,t,u_{\varepsilon},\nabla u_{\varepsilon})\cdot\nabla
u_{\varepsilon}\geq\frac{c_{1}m}{2}u_{\varepsilon}^{m-1}|\nabla
u_{\varepsilon}|^{2}-c_{3},$ $|a(x,t,u_{\varepsilon},\nabla
u_{\varepsilon})|\leq mc_{4}u_{\varepsilon}^{m-1}|\nabla
u_{\varepsilon}|+c_{4}|u_{\varepsilon}|^{\frac{m-1}{2}}$

and

$|b(x,t,u_{\varepsilon},\nabla u_{\varepsilon})|\leq c_{5}$

with some constants $c_{i}>0$ $(i=3,4,5)$ independent of $\varepsilon>0$.
Therefore as an application of the known result on Hölder regularity in scalar
parabolic equations ([25]), there exist $\kappa_{1}\in(0,1)$ and $C>0$ such
that for all $t>t_{0}$ and $\varepsilon\in(0,1)$,

$\|u_{\varepsilon}\|_{C^{\kappa_{1},\frac{\kappa_{1}}{2}}(\overline{\Omega}\times[t,t+1])}\leq
C,$

which along with (2.7) readily entails (5.1) with $\kappa=\kappa_{1}$.
Similarly one can also conclude that there exist $\kappa_{2}\in(0,1)$ and
$C>0$ such that

$\|v\|_{C^{\kappa_{2},\frac{\kappa_{2}}{2}}(\overline{\Omega}\times[t,t+1])}+\|w\|_{C^{\kappa_{2},\frac{\kappa_{2}}{2}}(\overline{\Omega}\times[t,t+1])}\leq
C.$ (5.4)

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

$\|uf(w)\|_{C^{\kappa_{3},\frac{\kappa_{3}}{2}}(\overline{\Omega}\times[t,t+1])}\leq
C$

with $\kappa_{3}=\min\\{\kappa_{1},\kappa_{2}\\}$. Thereupon (5.2) with
$\kappa=\kappa_{3}$ follows from the standard parabolic Schauder theory
([19]). ∎

The core of our proof of the stabilization result in Theorem 1.1 consists in
the following observation.

######  Lemma 5.2.

Assume that $m>1$ and $D\geq 1$, we have

$\int^{\infty}_{0}\int_{\Omega}uf(w)<\infty$ (5.5)

and

$\int^{\infty}_{0}\int_{\Omega}|\nabla w|^{2}<\infty.$ (5.6)

###### Proof.

An integration of the third equation in (2.19) yields

$\int_{\Omega}w_{\varepsilon}(\cdot,t)+\int^{t}_{0}\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})=\int_{\Omega}w_{0}\quad\hbox{for
all}~{}~{}t>0.$

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

$\int^{\infty}_{0}\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})\leq\int_{\Omega}w_{0}$
(5.7)

which implies (5.5) on an application of Fatou’s lemma, because
$u_{\varepsilon}f(w_{\varepsilon})\rightarrow uf(w)$ a.e. in
$\Omega\times(0,\infty)$.

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

$\frac{1}{2}\int_{\Omega}w^{2}_{\varepsilon}(\cdot,t)+\int^{t}_{0}\int_{\Omega}|\nabla
w_{\varepsilon}|^{2}=\frac{1}{2}\int_{\Omega}w^{2}_{0}-\int^{t}_{0}\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})w_{\varepsilon}\leq\frac{1}{2}\int_{\Omega}w^{2}_{0}$

and thereby verifies (5.6) via (2.12). ∎

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 5.3.

Let $m>1$ and $\alpha>0$. There exists $D_{0}\geq 1$ such that whenever
$D>D_{0}$, the solution of (1.10)–(1.12) constructed in Lemma 2.1 satisfies

$\int^{\infty}_{3}\int_{\Omega}|\nabla u^{\frac{m+1}{2}}|^{2}<\infty$ (5.8)

as well as

$\int^{\infty}_{3}\int_{\Omega}|\nabla v|^{2}<\infty$ (5.9)

###### Proof.

Testing the first equation of (2.5) by $(u_{\varepsilon}+\varepsilon)$ and
applying Young’s inequality, we obtain

$\displaystyle\frac{d}{dt}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2}+\int_{\Omega}v_{\varepsilon}^{-\alpha}(u_{\varepsilon}+\varepsilon)^{m-1}|\nabla
u_{\varepsilon}|^{2}$
$\displaystyle\leq\alpha^{2}\int_{\Omega}v_{\varepsilon}^{-\alpha-2}(u_{\varepsilon}+\varepsilon)^{m+1}|\nabla
v_{\varepsilon}|^{2}+2\int_{\Omega}(u_{\varepsilon}+\varepsilon)u_{\varepsilon}f(w_{\varepsilon}).$
(5.10)

On the other hand, let
$\mu_{\varepsilon}(t)=\left(\frac{1}{|\Omega|}\int_{\Omega}u^{\frac{m+1}{2}}_{\varepsilon}(\cdot,t)\right)^{\frac{2}{m+1}}$,
then testing the second equation of (2.5) by $-\Delta v_{\varepsilon}$ shows

$\displaystyle\frac{d}{dt}\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}+D\int_{\Omega}(\Delta
v_{\varepsilon})^{2}+2\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\leq\frac{1}{D}\int_{\Omega}|u_{\varepsilon}(\cdot,t)-\mu_{\varepsilon}(t)|^{2}.$
(5.11)

Hence combining (5.1) and (5.11), we obtain that for any $\eta>0$

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2}+\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\right)+\eta D\int_{\Omega}|\Delta
v_{\varepsilon}|^{2}+2\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}+\int_{\Omega}v^{-\alpha}_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}|\nabla
u_{\varepsilon}|^{2}\\\\[8.53581pt]
\leq&\displaystyle\frac{\eta}{D}\int_{\Omega}|u_{\varepsilon}(\cdot,t)-\mu_{\varepsilon}(t)|^{2}+\alpha^{2}\int_{\Omega}v_{\varepsilon}^{-\alpha-2}(u_{\varepsilon}+\varepsilon)^{m+1}|\nabla
v_{\varepsilon}|^{2}+2\int_{\Omega}(u_{\varepsilon}+\varepsilon)u_{\varepsilon}f(w_{\varepsilon}).\end{array}$
(5.12)

In view of Lemma 2.3 and Lemma 2.4, there exist $c_{i}>0(i=1,2)$ independent
of $D\geq 1$ satisfying

$v_{\varepsilon}^{-\alpha}(x,t)\geq c_{1},\quad
v_{\varepsilon}^{-\alpha-2}(x,t)\leq
c_{2}~{}~{}\hbox{in}~{}~{}\Omega\times(2,\infty)$ (5.13)

for all $\varepsilon\in(0,1)$. Therefore from (5.12), it follows that

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2}+\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\right)+\eta D\int_{\Omega}|\Delta
v_{\varepsilon}|^{2}+2\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}+c_{1}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m-1}|\nabla
u_{\varepsilon}|^{2}\\\\[8.53581pt]
\leq&\displaystyle\frac{\eta}{D}\int_{\Omega}|u_{\varepsilon}(\cdot,t)-\mu_{\varepsilon}(t)|^{2}+\alpha^{2}c_{2}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+1}|\nabla
v_{\varepsilon}|^{2}+2\int_{\Omega}(u_{\varepsilon}+\varepsilon)u_{\varepsilon}f(w_{\varepsilon}).\end{array}$
(5.14)

According to Lemma 4.3 with $p=2(m+1)$, we have

$\displaystyle\left(\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2(m+1)}\right)^{\frac{1}{2}}\leq
c_{3},$

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

$\begin{array}[]{rl}&\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+1}|\nabla
v_{\varepsilon}|^{2}\\\
\leq&\displaystyle\left(\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2(m+1)}\right)^{\frac{1}{2}}\left(\int_{\Omega}|\nabla
v_{\varepsilon}|^{4}\right)^{\frac{1}{2}}\\\
\leq&c_{4}\displaystyle\left(\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2(m+1)}\right)^{\frac{1}{2}}\left(\|\Delta
v_{\varepsilon}\|^{2}+\|\nabla v_{\varepsilon}\|^{2}_{L^{2}(\Omega)}\right)\\\
\leq&c_{3}c_{4}(\|\Delta v_{\varepsilon}\|^{2}+\|\nabla
v_{\varepsilon}\|^{2}_{L^{2}(\Omega)}).\end{array}$ (5.15)

Therefore inserting (5.15) into (5.14) yields

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}(\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2}+\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2})+\eta D\int_{\Omega}|\Delta
v_{\varepsilon}|^{2}+2\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}+\frac{4c_{1}}{(m+1)^{2}}\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}\\\\[8.53581pt]
\leq&\displaystyle\frac{\eta}{D}\int_{\Omega}|u_{\varepsilon}(\cdot,t)-\mu_{\varepsilon}(t)|^{2}+\alpha^{2}c_{2}c_{3}c_{4}(\|\Delta
v_{\varepsilon}\|^{2}_{L^{2}(\Omega)}+\|\nabla
v_{\varepsilon}\|^{2}_{L^{2}(\Omega)})+2\int_{\Omega}(u_{\varepsilon}+\varepsilon)u_{\varepsilon}f(w_{\varepsilon}).\end{array}$
(5.16)

By the elementary inequality:

$\frac{\xi^{\mu}-\eta^{\mu}}{\xi-\eta}\geq\eta^{\mu-1}~{}~{}\hbox{for}~{}~{}\mu\geq
1,\xi\geq 0,\eta\geq 0~{}~{}\hbox{ and}~{}~{}\xi\neq\eta,$

we have

$|u_{\varepsilon}^{\frac{m+1}{2}}(\cdot,t)-\mu_{\varepsilon}^{\frac{m+1}{2}}|\geq\mu_{\varepsilon}(\cdot,t)^{\frac{m-1}{2}}|u_{\varepsilon}(\cdot,t)-\mu_{\varepsilon}(t)|$

and thus

${\mu}^{m-1}_{\varepsilon}(t)\int_{\Omega}|u_{\varepsilon}(\cdot,t)-\mu_{\varepsilon}(t)|^{2}\leq\int_{\Omega}|u^{\frac{m+1}{2}}_{\varepsilon}(\cdot,t)-\mu^{\frac{m+1}{2}}_{\varepsilon}(t)|^{2}.$
(5.17)

Furthermore by the Hölder inequality and the noncreasing property of
$t\mapsto\int_{\Omega}u_{\varepsilon}(\cdot,t)$,

$\mu_{\varepsilon}(t)\geq\frac{1}{|\Omega|}\int_{\Omega}u_{\varepsilon}(\cdot,t)\geq\frac{1}{|\Omega|}\int_{\Omega}u_{0}$

and thereby the Poincaré inequality entails that for some $c_{5}>0$

$\begin{array}[]{rl}&\overline{u_{0}}^{m-1}\displaystyle\int_{\Omega}|u_{\varepsilon}(\cdot,t)-\mu_{\varepsilon}(t)|^{2}\\\\[8.53581pt]
\leq&\displaystyle\int_{\Omega}|u^{\frac{m+1}{2}}_{\varepsilon}(\cdot,t)-\mu^{\frac{m+1}{2}}_{\varepsilon}(t)|^{2}\\\\[5.69054pt]
\leq&c_{5}\displaystyle\int_{\Omega}|\nabla
u^{\frac{m+1}{2}}_{\varepsilon}|^{2}\\\\[5.69054pt]
\leq&c_{5}\displaystyle\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}.\end{array}$
(5.18)

Hence substituting (5.18) into (5.16) shows that

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}(\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{2}+\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2})+(\frac{4c_{1}}{(m+1)^{2}}-\frac{\eta
c_{5}}{D\overline{u_{0}}^{m-1}})\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}\\\\[8.53581pt]
\leq&\displaystyle(\alpha^{2}c_{2}c_{3}c_{4}-\eta D)\|\Delta
v_{\varepsilon}\|^{2}_{L^{2}(\Omega)}+(\alpha^{2}c_{2}c_{3}c_{4}-2\eta)\|\nabla
v_{\varepsilon}\|^{2}_{L^{2}(\Omega)}+2\displaystyle\int_{\Omega}(u_{\varepsilon}+\varepsilon)u_{\varepsilon}f(w_{\varepsilon})\\\
\leq&\displaystyle(\alpha^{2}c_{2}c_{3}c_{4}-\eta)\|\Delta
v_{\varepsilon}\|^{2}_{L^{2}(\Omega)}+(\alpha^{2}c_{2}c_{3}c_{4}-2\eta)\|\nabla
v_{\varepsilon}\|^{2}_{L^{2}(\Omega)}+2\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+1)\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})\end{array}$

and hence completes the proof upon the choice of
$D_{0}:=\max\\{1,\frac{\alpha^{2}c_{2}c_{3}c_{4}c_{5}(m+1)^{2}}{3c_{1}\overline{u_{0}}^{m-1}}\\}$.
Indeed, for any $D>D_{0}$, it is possible to find $\eta>0$ such that

$\frac{3c_{1}}{(m+1)^{2}}\geq\frac{\eta
c_{5}}{D\overline{u_{0}}^{m-1}},~{}~{}\alpha^{2}c_{2}c_{3}c_{4}\leq\eta$

and thereby

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}(\int_{\Omega}|u_{\varepsilon}+\varepsilon|^{2}+\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2})+\frac{c_{1}}{(m+1)^{2}}\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}+\eta\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\\\\[8.53581pt]
\leq&2(\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+1)\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon}).\end{array}$

Therefore, in view of (5.7), (4.17) and (4.18) we see that for any $t>3$,

$\displaystyle\int^{t}_{3}\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}+\displaystyle\int^{t}_{3}\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\leq
c_{6}+c_{6}\displaystyle\int^{\infty}_{3}\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})\leq
c_{6}+c_{6}\int_{\Omega}w_{0}.$ (5.19)

with constant $c_{6}>0$ independent of $\varepsilon$ and time $t$, which
implies that (5.8) and (5.9) is valid due to the lower semicontinuity of
norms. ∎

###  5.2 Decay of $w$

The integrability statement in Lemma 5.2 can be turned into the decay property
of $w$ with respect to the norm in $L^{\infty}(\Omega)$, thanks to the fact
that $\|u(\cdot,t)\|_{L^{1}(\Omega)}$ is increasing with time, while
$\|w(\cdot,t)\|_{L^{\infty}(\Omega)}$ is nonincreasing.

######  Lemma 5.4.

The third component of the weak solution of (1.10)–(1.12) constructed in Lemma
2.1 fulfills

$\|w(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow 0\quad\textnormal{\emph{as
$t\rightarrow\infty$}}.$ (5.20)

###### Proof.

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

$\displaystyle\overline{u_{0}}\cdot\int_{\Omega}f(w)$ $\displaystyle=$
$\displaystyle\int_{\Omega}u\overline{f(w)}$ $\displaystyle=$
$\displaystyle\int_{\Omega}uf(w)-\int_{\Omega}u(f(w)-\overline{f(w)})$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}uf(w)+c_{1}\|u\|_{L^{\infty}(\Omega)}\|f^{\prime}(w)\|_{L^{\infty}(\Omega)}\left\\{\int_{\Omega}|\nabla
w|^{2}\right\\}^{\frac{1}{2}}.$

Thanks to the boundedness of $u$ and $w$, we have

$\displaystyle{\overline{u_{0}}}^{2}\cdot\left\\{\int_{\Omega}f(w)\right\\}^{2}$
$\displaystyle\leq$ $\displaystyle
2\left\\{\int_{\Omega}uf(w)\right\\}^{2}+c_{2}\int_{\Omega}|\nabla w|^{2}$
$\displaystyle\leq$ $\displaystyle
c_{3}\int_{\Omega}uf(w)+c_{2}\int_{\Omega}|\nabla w|^{2}.$

Hence from Lemma 5.2 it follows that

$\int^{\infty}_{1}\|f(w(\cdot,t))\|^{2}_{L^{1}(\Omega)}dt<\infty.$

which, along with the uniformly Hölder estimate from Lemma 5.1, implies that

$f(w(\cdot,t))\rightarrow 0\quad\textnormal{in
$L^{1}(\Omega)$}\quad\textnormal{as $t\rightarrow\infty$}$ (5.21)

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 $\Omega$. 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 $\Omega$ as $t_{j}\rightarrow\infty.$ Furthermore, the dominated
convergence theorem ensures that

$w(\cdot,t_{j})\rightarrow 0\quad\textnormal{in
$L^{1}(\Omega)$}\quad\textnormal{as $t_{j}\rightarrow\infty$}.$ (5.22)

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

$\|w(\cdot,t_{j})\|_{L^{\infty}(\Omega)}\leq c_{4}\|\nabla
w(\cdot,t_{j})\|^{\frac{4}{5}}_{L^{4}(\Omega)}\|w(\cdot,t_{j})\|^{\frac{1}{5}}_{L^{1}(\Omega)}+c_{4}\|w(\cdot,t_{j})\|_{L^{1}(\Omega)}$

and thus

$\|w(\cdot,t_{j})\|_{L^{\infty}(\Omega)}\rightarrow 0\quad\textnormal{as
$t_{j}\rightarrow\infty$}.$ (5.23)

Since $t\mapsto\|w(\cdot,t)\|_{L^{\infty}(\Omega)}$ is noncreasing by Lemma
2.3, (5.20) indeed results from (5.23). ∎

###  5.3 Convergence of $u$

In this subsection, 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
$\int^{\infty}_{3}\int_{\Omega}|\nabla u^{\frac{m+1}{2}}|^{2}$ in Lemma 5.3,
which implies that $\|\nabla u^{\frac{m+1}{2}}(\cdot,t_{k})\|_{L^{2}(\Omega)}$
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 5.5.

We have

$\int^{\infty}_{3}\|u_{t}(\cdot,t)\|^{2}_{(W_{0}^{1,2}(\Omega))^{*}}dt<\infty.$
(5.24)

###### Proof.

For any $\varphi\in C_{0}^{\infty}(\Omega)$, multiplying the first equation in
(2.5) by $\varphi$ and integrating by parts over $\Omega$ yields

$\begin{array}[]{rl}|\displaystyle\int_{\Omega}u_{\varepsilon
t}\varphi|&=\left|\displaystyle\int_{\Omega}\varepsilon\nabla(u_{\varepsilon}+1)^{M}\cdot\nabla\varphi+\nabla(u_{\varepsilon}(u_{\varepsilon}+\varepsilon)^{m-1}v_{\varepsilon}^{-\alpha})\cdot\nabla\varphi+\beta
u_{\varepsilon}f(w_{\varepsilon})\varphi\right|\\\\[8.53581pt]
&\leq\displaystyle\int_{\Omega}(M(u_{\varepsilon}+1)^{M-1}|\nabla
u_{\varepsilon}|+mv_{\varepsilon}^{-\alpha}(u_{\varepsilon}+\varepsilon)^{m-1}|\nabla
u_{\varepsilon}|+\alpha(u_{\varepsilon}+1)^{m}v_{\varepsilon}^{-\alpha-1}|\nabla
v_{\varepsilon}|)|\nabla\varphi|\\\\[8.53581pt]
&+\displaystyle\beta\int_{\Omega}|u_{\varepsilon}f(w_{\varepsilon})|\|\varphi\|_{L^{\infty}(\Omega)}\\\
&\leq
c_{1}(\left\\{\displaystyle\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}\right\\}^{\frac{1}{2}}+\displaystyle\left\\{\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\right\\}^{\frac{1}{2}})\|\varphi\|_{W^{1,2}(\Omega)}+\beta\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})\|\varphi\|_{L^{\infty}(\Omega)}\end{array}$

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
$W^{1,2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$, the above inequality
implies that

$\|u_{\varepsilon t}(\cdot,t)\|_{(W_{0}^{1,2}(\Omega))^{*}}\leq
c_{1}(\left\\{\displaystyle\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}\right\\}^{\frac{1}{2}}+\displaystyle\left\\{\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}\right\\}^{\frac{1}{2}})+\beta\displaystyle\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})$

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

$\displaystyle\int^{T}_{3}\|u_{\varepsilon
t}(\cdot,t)\|^{2}_{(W^{1,2}(\Omega))^{*}}dt\leq
c_{2}\left(\int^{T}_{3}\displaystyle\int_{\Omega}|\nabla(u_{\varepsilon}+\varepsilon)^{\frac{m+1}{2}}|^{2}+\displaystyle\int^{T}_{3}\int_{\Omega}|\nabla
v_{\varepsilon}|^{2}+\displaystyle\int^{T}_{3}\int_{\Omega}u_{\varepsilon}f(w_{\varepsilon})\right)$

which together with (5.19) leads to

$\int^{\infty}_{3}\|u_{\varepsilon
t}(\cdot,t)\|^{2}_{(W_{0}^{1,2}(\Omega))^{*}}dt\leq c_{3}$

with $c_{3}>0$ independent of $\varepsilon$. Hence (5.24) results from lower
semi-continuity of the norm in the Hilbert space
$L^{2}((3,\infty);(W_{0}^{1,2}(\Omega))^{*})$ with respect to weak
convergence. ∎

Thanks to above estimates, we adapt the argument in [36] to show that $u$
actually stabilizes toward $\overline{u_{0}}+\beta\overline{w_{0}}$ in the
claimed sense beyond in the weak-$*$ sense in $L^{\infty}(\Omega)$.

######  Lemma 5.6.

Let $m>1,\alpha>0$ and suppose that $D\geq D_{0}$. Then we have

$\|u(\cdot,t)-u_{\star}\|_{L^{\infty}(\Omega)}\rightarrow
0\quad\textnormal{\emph{as}}~{}~{}t\rightarrow\infty,$ (5.25)

where
$u_{\star}=\frac{1}{|\Omega|}\int_{\Omega}u_{0}+\frac{\beta}{|\Omega|}\int_{\Omega}w_{0}$.

###### Proof.

According to Lemma 5.3 and Lemma 5.5, one can conclude that

$u(\cdot,t)\stackrel{{\scriptstyle\mathrm{w}^{*}}}{{\rightharpoonup}}u_{\star}\quad\hbox{in}~{}L^{\infty}(\Omega)~{}~{}\textnormal{\emph{as}}~{}~{}t\rightarrow\infty$
(5.26)

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 $\tilde{\psi}\in L^{1}(\Omega)$ such that

$\int_{\Omega}u(x,t_{k})\tilde{\psi}dx-\int_{\Omega}u_{\star}\tilde{\psi}dx\geq
c_{1}~{}\hbox{for all}~{}k\in\mathbb{N}$

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

$\int_{\Omega}u(x,t_{k})\psi dx-\int_{\Omega}u_{\star}\psi
dx\geq\frac{3c_{1}}{4}~{}~{}~{}\hbox{for all}~{}k\in\mathbb{N}.$

and then

$\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}u(x,t)\psi
dxdt-\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}u_{\star}\psi
dxdt\geq\frac{c_{1}}{2}~{}~{}~{}\hbox{for all sufficently large
}~{}k\in\mathbb{N},$ (5.27)

where we have used the fact that

$\begin{array}[]{rl}&\left|\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}(u(x,t)-u(x,t_{k}))\psi
dx\right|\\\\[5.69054pt]
=&\left|\displaystyle\int^{t_{k}+1}_{t_{k}}\int^{t}_{t_{k}}\langle
u_{t}(\cdot,s),\psi(\cdot)\rangle dsdt\right|\\\
\leq&\displaystyle\int^{t_{k}+1}_{t_{k}}\int^{t}_{t_{k}}\|u_{t}(\cdot,s)\|_{(W^{1,2}_{0}(\Omega))^{*}}dsdt\cdot\|\psi\|_{W^{1,2}_{0}(\Omega)}\\\\[11.38109pt]
\leq&\displaystyle\int^{t_{k}+1}_{t_{k}}\left\\{\int^{t}_{t_{k}}\|u_{t}(\cdot,s)\|^{2}_{(W^{1,2}_{0}(\Omega))^{*}}ds\right\\}^{\frac{1}{2}}|t-t_{k}|^{\frac{1}{2}}dt\cdot\|\psi\|_{W^{1,2}_{0}(\Omega)}\\\\[11.38109pt]
\leq&\left\\{\displaystyle\int^{t_{k}+1}_{t_{k}}\int^{t}_{t_{k}}\|u_{t}(\cdot,s)\|^{2}_{(W^{1,2}_{0}(\Omega))^{*}}dsdt\right\\}^{\frac{1}{2}}\cdot\|\psi\|_{W^{1,2}_{0}(\Omega)}\\\\[11.38109pt]
\leq&\left\\{\displaystyle\int^{\infty}_{t_{k}}\|u_{t}(\cdot,s)\|^{2}_{(W^{1,2}_{0}(\Omega))^{*}}ds\right\\}^{\frac{1}{2}}\cdot\|\psi\|_{W^{1,2}_{0}(\Omega)}\\\
\longrightarrow&0~{}~{}\hbox{as}~{}~{}k\rightarrow\infty,\end{array}$

due to Lemma 5.5.

Let
$\mu(t)=\left(\frac{1}{|\Omega|}\int_{\Omega}u^{\frac{m+1}{2}}(\cdot,t)\right)^{\frac{2}{m+1}}$.
Then as in (5.18), we have

$\overline{u_{0}}^{m-1}\int_{\Omega}|u(\cdot,t)-\mu(t)|^{2}\leq\int_{\Omega}|u^{\frac{m+1}{2}}(\cdot,t)-\mu^{\frac{m+1}{2}}(t)|^{2}\leq
c_{5}\int_{\Omega}|\nabla u^{\frac{m+1}{2}}|^{2}$

and thus

$\overline{u_{0}}^{m-1}\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}|u(\cdot,t)-\mu(t)|^{2}\leq
c_{5}\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}|\nabla
u^{\frac{m+1}{2}}(\cdot,t)|^{2}.$ (5.28)

We now introduce

$u_{k}(x,s):=u(x,t_{k}+s),(x,s)\in\Omega\times(0,1)$

and

$\mu_{k}(x,s):=\mu(x,t_{k}+s),(x,s)\in\Omega\times(0,1)$

for $k\in\mathbb{N}$. Then (5.28) implies that

$\begin{array}[]{rl}\overline{u_{0}}^{m-1}\displaystyle\int^{1}_{0}\int_{\Omega}|u_{k}(\cdot,s)-\mu_{k}(s)|^{2}ds\leq&c_{5}\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}|\nabla
u^{\frac{m+1}{2}}(\cdot,t)|^{2}\\\ &\longrightarrow 0~{}~{}\hbox{as
}~{}~{}k\rightarrow\infty,\end{array}$

due to (5.8) in Lemma 5.3. This means that

$u_{k}(x,s)-\mu_{k}(s)\rightarrow
0~{}~{}\hbox{in}~{}~{}L^{2}(\Omega\times(0,1))~{}~{}\hbox{as}~{}~{}k\rightarrow\infty,$
(5.29)

which in particular allows us to get

$\displaystyle\int^{1}_{0}\int_{\Omega}(u_{k}(\cdot,s)-\mu_{k}(s))\psi(\cdot)ds\rightarrow
0~{}\hbox{as}~{}~{}k\rightarrow\infty$ (5.30)

as well as

$\displaystyle\int^{1}_{0}\int_{\Omega}(u_{k}(\cdot,s)-\mu_{k}(s))ds\rightarrow
0~{}\hbox{as}~{}~{}k\rightarrow\infty.$ (5.31)

thanks to the weak convergence of $L^{2}(\Omega\times(0,1))$. Moreover, by
Lemma 5.4, we have

$\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}w(\cdot,t)dt\leq|\Omega|\|w(\cdot,t_{k})\|_{L^{\infty}(\Omega)}\rightarrow
0~{}\hbox{as}~{}~{}k\rightarrow\infty$

and thereby

$\begin{array}[]{rl}|\Omega|\displaystyle\int^{1}_{0}\mu_{k}(s)ds=&\displaystyle\int^{1}_{0}\int_{\Omega}u_{k}(\cdot,s)ds-\displaystyle\int^{1}_{0}\int_{\Omega}(u_{k}(\cdot,s)-\mu_{k}(s))ds\\\
=&|\Omega|u_{*}-\beta\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}w(\cdot,t)dt-\displaystyle\int^{1}_{0}\int_{\Omega}(u_{k}(\cdot,s)-\mu_{k}(s))ds\\\
\rightarrow&|\Omega|u_{*}~{}\hbox{as}~{}~{}k\rightarrow\infty\end{array}$
(5.32)

due to (5.31) and (5.20).

Therefore from (5.27), (5.30) and (5.32), it follows that

$\begin{array}[]{rl}\displaystyle\frac{c_{1}}{2}\leq&\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}u(\cdot,t)\psi(\cdot)dt-\displaystyle\int^{t_{k}+1}_{t_{k}}\int_{\Omega}u_{\star}\psi(\cdot)dt\\\\[11.38109pt]
=&\displaystyle\int^{1}_{0}\int_{\Omega}(u_{k}(\cdot,s)-\mu_{k}(s))\psi(\cdot)ds+\int^{1}_{0}\int_{\Omega}\mu_{k}(s)\psi(\cdot)ds-\displaystyle
u_{\star}\int_{\Omega}\psi(\cdot)\\\\[8.53581pt]
=&\displaystyle\int^{1}_{0}\int_{\Omega}(u_{k}(\cdot,s)-\mu_{k}(s))\psi(\cdot)ds+\int^{1}_{0}\mu_{k}(s)ds\int_{\Omega}\psi(\cdot)-u_{\star}\displaystyle\int_{\Omega}\psi(\cdot)\\\
\rightarrow&0~{}\hbox{as}~{}~{}k\rightarrow\infty,\end{array}$

which is absurd and hence proves that actually (5.26) is valid.

Let us suppose on the contrary that (5.25) 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 $c_{1}>0$

$u(x_{k},t_{k})-u_{*}=\max_{x\in\Omega}|u(x,t_{k})-u_{*}|\geq
c_{1}~{}~{}\hbox{for all}~{}k\in\mathbb{N}.$

In view of the compactness of $\overline{\Omega}$, where passing to
subsequences we can find $x_{0}\in\overline{\Omega}$ such that
$x_{k}\rightarrow x_{0}$ as $k\rightarrow\infty$. Furthermore, because $u$ is
uniformly continuous in $\bigcup_{~{}k\in\mathbb{N}}(\overline{\Omega}\times
t_{k})$, this entails that one can extract a further subsequence if necessary
such that

$u(x,t_{k})-u_{*}\geq\frac{c_{1}}{2}~{}~{}~{}\hbox{for all}~{}x\in
B:=B_{\delta}(x_{0})\cap\Omega~{}\hbox{and}~{}k\in\mathbb{N}$

for some $\delta>0$. Noticing that if $x_{0}\in\partial\Omega$, the smoothness
of $\partial\Omega$ ensures the existence of $\hat{x}_{0}\in\Omega$ 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

$\int_{\Omega}(u(x,t_{k})-u_{\star})\psi
dx=\int_{B_{\hat{\delta}}(\hat{x}_{0})}(u(x,t_{k})-u_{\star})\psi
dx\geq\frac{c_{1}}{2}\cdot\int_{\Omega}\psi dx,$

which contradicts (5.26) and hence proves the lemma. ∎

###  5.4 Stabilization of $v$

In what follows, based on the uniform Hölder bounds of $v$ and decay of
$\nabla v$ implied by (5.2) and (5.9) respectively, we shall show the
corresponding stabilization result for $v$ by a contradiction argument.

######  Lemma 5.7.

Let $m>1$ and $(u,v,w)$ be the solution of (1.10)–(1.12) obtained in Lemma
2.1. Then we have

$\|v(\cdot,t)-u_{\star}\|_{L^{\infty}(\Omega)}\rightarrow
0\quad\textnormal{\emph{as $t\rightarrow\infty$}}.$ (5.33)

###### Proof.

Combined the uniform Hölder bounds of $v$ and decay of $\nabla v$ implied by
(5.2) and (5.9), respectively, (5.33) may be derived by a contradiction
argument. Indeed, assume that (5.33) 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

$\|v(\cdot,t_{k})-u_{\star}\|_{L^{\infty}}\geq c_{1}.$

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

$|v(x,t)-u_{\star}|>\frac{c_{1}}{2}$

for every $x\in B_{r}(x_{k})$ and $t\in(t_{k},t_{k}+\tau)$ and hence

$\int^{t_{k}+\tau}_{t_{k}}\int_{\Omega}|v(\cdot,t)-u_{\star}|^{2}>\frac{|\Omega|\tau
c^{2}_{1}}{4}.$ (5.34)

On the other hand, the Poincaré inequality indicates

$\int^{t_{k}+\tau}_{t_{k}}\int_{\Omega}|v(\cdot,t)-u_{\star}|^{2}\leq
C\int^{t_{k}+\tau}_{t_{k}}\int_{\Omega}|\nabla
v|^{2}+C\int^{t_{k}+\tau}_{t_{k}}\int_{\Omega}|\overline{v(\cdot,t)}-u_{\star}|^{2}.$
(5.35)

Therefore (5.35) yields a contradiction to (5.34) thanks to

$\int^{t_{k}+\tau}_{t_{k}}\int_{\Omega}|\nabla v|^{2}\rightarrow
0\quad\textnormal{as $t_{k}\rightarrow\infty$}.$

∎

Now the convergence result in the flavor of Theorem 1.1 has actually been
proved already.

Proof of Theorem 1.1 The claimed assertion in Theorem 1.1 is the consequence
of Lemma 5.4, 5.6 and 5.7.

##  6 Acknowledgments

This work is supported by the NNSF of China (No.12071030) and Beijing key
laboratory on MCAACI.

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