\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 176, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/176\hfil Boundedness in a chemotaxis system]
{Boundedness in a three-dimensional attraction-repulsion
chemotaxis system with nonlinear diffusion and logistic source}

\author[Y. Wang \hfil EJDE-2016/176\hfilneg]
{Yilong Wang}

\address{Yilong Wang \newline
 School of Sciences,
 Southwest Petroleum University,
 Chengdu 610500, China}
\email{wangelongelone@163.com}

\thanks{Submitted February 8, 2016. Published July 6, 2016.}
\subjclass[2010]{35K55, 35Q35, 35Q92, 92C17}
\keywords{Chemotaxis; attraction-repulsion; boundedness; nonlinear diffusion;
\hfill\break\indent logistic source}

\begin{abstract}
 This article concerns the attraction-repulsion chemotaxis system with
 nonlinear diffusion and logistic source,
 \begin{gather*}
 u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla\cdot(\chi u\nabla v)
 +\nabla\cdot(\xi u\nabla w)+ru-\mu u^\eta, \\
 x\in\Omega,\; t>0,\\
 v_t=\Delta v+\alpha u-\beta v, \quad x\in\Omega, \; t>0,\\
 w_t=\Delta w+\gamma u-\delta w, \quad x\in\Omega,\; t>0
 \end{gather*}
 under Neumann boundary conditions in a bounded domain
 $\Omega\subset\mathbb{R}^3$ with smooth boundary.
 We show that if the diffusion is strong enough or
 the logistic dampening is sufficiently powerful, then the corresponding
 system possesses a global bounded classical solution for any sufficiently
 regular initial data. Moreover, it is proved that if $r=0$,
 $\beta>\frac{1}{2(\eta-1)}$ and $\delta>\frac{1}{2(\eta-1)}$ for the latter case,
 then $u(\cdot,t)\to 0$, $ v(\cdot,t)\to 0$ and $ w(\cdot,t)\to 0$ in
 $L^\infty(\Omega)$ as $ t \to \infty$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


 \section{Introduction}

 We consider the attraction-repulsion chemotaxis system with nonlinear
diffusion and logistic source,
 \begin{equation}\label{I1}
\begin{gathered}
u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla\cdot(\chi u\nabla v)
 +\nabla\cdot(\xi u\nabla w)+ru-\mu u^\eta, \\
 x\in\Omega,\; t>0,\\
v_t=\Delta v+\alpha u-\beta v, \quad x\in\Omega, \; t>0,\\
w_t=\Delta w+\gamma u-\delta w,\quad x\in\Omega,\; t>0,\\
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}
=\frac{\partial w}{\partial \nu}=0,\quad x\in\partial\Omega,\; t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad w(x,0)=w_0(x),\quad x\in\Omega,
\end{gathered}
\end{equation}
in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary,
where $\frac{\partial}{\partial \nu}$ denotes the outward normal derivative
 on $\partial \Omega$, $m\geq1$, $r\geq 0$, $\mu>0$, $\eta>1$, $\chi>0$,
 $\xi>0$, $\alpha>0$, $\beta>0$, $\gamma >0$ and $\delta>0$ are prescribed
parameters. Here $u(x,t)$ denotes the cell density, $v(x,t)$ and $w(x,t)$
represent the chemoattractant concentration and the chemorepellent
concentration, respectively. In \eqref{I1} we assume that cell kinetics
follows a logistic-type law determined by parameters $r$, $\mu$ and $\eta$.
The first cross-diffusive term and the second in the first equation reflect
the attractive migration and repulsive movement of cells, respectively.
The second and the third equations imply that the attractive and repulsive
signals are produced by cells themselves. System \eqref{I1} is a generalized
version of the classical Keller-Segel model \cite{KS} and can describe
the aggregation of microglia in Alzheimer's disease due to the interaction
of chemoattractant and chemorepellent \cite{LCE}.
System \eqref{I1} can also address the quorum effect in the chemotactic
process \cite{PH}. The mathematical studies on system \eqref{I1} are much
harder than the classical Keller-Segel system due to lacking a useful
Lyapunov functional.

To motivate our study, we recall some related works on system \eqref{I1}.
In the case $m=1$, $r=0$ and $\mu=0$, there have been a series of development:
Wang \emph{et al.} \cite{LW,JW} proved that system \eqref{I1} possesses
a unique global bounded solution when $n=1$; when $n=2$ and the repulsion
prevails over the attraction in the sense that $ \xi\gamma-\chi\alpha>0$,
Jin \cite{Jin} and Tao \emph{et al.} \cite{LiuTao} independently proved
that system \eqref{I1} admits a unique global bounded solution;
however, when $n=3$ and $ \xi\gamma-\chi\alpha>0$, the global existence
of classical solutions for \eqref{I1} is still open; Jin \cite{Jin}
only proved that system \eqref{I1} possesses a global weak solution;
when $n=2,3$ and the repulsion cancels the attraction
(i.e. $\xi\gamma=\chi\alpha$ ), the authors in \cite{LMW, JinLiu}
proved that system \eqref{I1} possesses a unique global bounded classical
solution and the large time behavior of solutions is considered for the
bounded domain and the whole space. The authors in \cite{LSW} investigated
the pattern formation of \eqref{I1} analytically and numerically for $n=1$.
For the parabolic-elliptic case, the global solvability, critical mass phenomenon,
blow-up, and large time behavior were investigated for the bounded domain
and the whole space (see \cite{ES,SW,TW,Li-Lin-Mu,JinWang}).
In \cite{Wang-BVP,Wang-DCDSB}, the global solvability and the uniform
boundedness were considered for the attraction-repulsion chemotaxis
system with nonlinear diffusion.


 In the case $m=1$, $r>0$ and $\mu>0$, there are only few results:
Li \emph{et al.} \cite{Li-Xiang2, Li-Xie} proved that \eqref{I1} with
$\eta\geq 1$ for $n=1$ or $\eta\geq 2$ for $n=2$ admits a unique
global bounded solution. For $n\geq 3$, however, the global existence
of classical solutions of \eqref{I1} is still open.
In \cite {Wang-JMAA}, the global solvability, boundedness and large
time behavior were only investigated for a quasilinear attraction-repulsion
chemotaxis model with logitic source for parabolic-elliptic type
in a bounded domain.

To the best of our knowledge, there is no rigorous result on the
quasilinear attraction-repulsion chemotaxis model with logistic source
for parabolic-parabolic type except \cite{Li-Xie}, where the author
only considered two-dimensional case and proved that when $m>1$
or $\eta\geq 2$, system \eqref{I1} possesses a global bounded classical solution.
Thus the goal of this paper is to explore the interactions between the
nonlinear diffusion and logistic source on the solutions of system \eqref{I1}
for three-dimensional case and partially answer the above open problem.
Throughout this paper, we assume that the initial data satisfy
\begin{equation}\label{I2}
\begin{gathered}
 u_0\in W^{1,\infty}(\Omega),\quad u_0>0\quad \text{on} \quad \bar{\Omega},\\
 v_0\in W^{1,\infty}(\Omega),\quad v_0\geq0\quad\, \text{on} \quad \bar{\Omega},\\
 w_0\in W^{1,\infty}(\Omega),\quad w_0\geq0\quad \text{on} \quad \bar{\Omega}.
 \end{gathered}
\end{equation}
We now state the main results of this article.

\begin{theorem}\label{thm1}
Let $\Omega\subset \mathbb{R}^{3}$ be a bounded domain with smooth boundary
and initial data $(u_0,v_0,w_0)$ satisfy \eqref{I2}.
Suppose that $m\geq1$, $r\geq 0$, $\mu>0$, $\eta>1$, $\chi>0$, $\xi>0$,
$\alpha>0$, $\beta>0$, $\gamma >0$ and $\delta>0$.
If one of the following two cases holds:
\begin{itemize}
\item[(i)] $\eta\geq 2$ and $\mu>\max\big\{1+4\chi^2+11\alpha^2
+2\chi^2(9+\alpha^2+\gamma^2)$, $1+4\xi^2+11\gamma^2
 +2\xi^2(9+\alpha^2+\gamma^2) \big\}$;

\item[(ii)] $\eta\in(1, 2)$ and $m>4/3$,
\end{itemize}
then there exists a triplet $ (u,v,w)$ belonging to
$ C^{0}\big(\bar{\Omega} \times[0,\infty)\big)\cap C^{2,1}\big(\bar{\Omega}
\times (0,\infty)\big)$ which solves system \eqref{I1} classically.
Moreover, $u$, $v$ and $w$ are bounded in $\Omega\times (0,\infty)$
in the sense that
 \begin{equation}\label{I3}
 \|u(\cdot,t)\|_{L^\infty(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}
+\|w(\cdot,t)\|_{W^{1,\infty}(\Omega)}
 \leq M \quad \forall t\in(0,\infty),
 \end{equation}
 where $M>0$ is a constant independent of $t$.
 \end{theorem}

\begin{remark} \label{rmk1.1} \rm
 Theorem \ref{thm1} answers the open problem in \cite[Remark 1.4]{Li-Xie}
and \cite[Remark 1.3]{Li-Xiang2} for three-dimensional case.
\end{remark}

\begin{remark} \label{rmk1.2} \rm
For the case $n\geq4$, if $m>2-\frac{2}{n}$, then Theorem \ref{thm1} holds.
This can be proved by the same arguments of case (ii) in Theorem \ref{thm1}.
\end{remark}

 Next, we consider the large time behavior of solutions to \eqref{I1}
for a special case $r=0$, $\eta\geq 2$,
$\mu>\max\big\{1+4\chi^2+11\alpha^2+2\chi^2(9+\alpha^2+\gamma^2)$,
$1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2) \big\}$,
$\beta>\frac{1}{2(\eta-1)}$ and $\delta>\frac{1}{2(\eta-1)}$
by using the ideas in \cite{T-W-ZAMP}. Here $r=0$ means that either
 cells are a prior unable to reproduce themselves, or the considered time
scales are much smaller than those of cell proliferation
 (see \cite{E-S-NA,K-R-CPDE} for details).

\begin{theorem}\label{thm2}
Let $\Omega\subset \mathbb{R}^{3}$ be a bounded domain with smooth
boundary and initial data $(u_0,v_0,w_0)$ satisfy \eqref{I2}.
Suppose that $m\geq1$, $\chi>0$, $\xi>0$, $\alpha>0$, $\beta>0$, $\gamma >0$
 and $\delta>0$. If $r=0$, $\eta\geq 2$,
$\mu>\max\big\{1+4\chi^2+11\alpha^2+2\chi^2(9+\alpha^2+\gamma^2)$,
$1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2) \big\}$,
$\beta>\frac{1}{2(\eta-1)}$ and $\delta>\frac{1}{2(\eta-1)}$,
then the global solution $(u,v,w)$ of system \eqref{I1}
satisfies
\begin{equation}\label{I4}
 \|u(\cdot, t)\|_{L^\infty(\Omega)}\to 0, \quad
\|v(\cdot, t)\|_{L^\infty(\Omega)}\to 0, \quad
\|w(\cdot, t)\|_{L^\infty(\Omega)}\to 0 \quad \text{as } t\to \infty.
\end{equation}
\end{theorem}

This article is organized as follows. In Section 2, we state the
 existence of local solutions to \eqref{I1} and some preliminary inequalities.
 In Section 3, we give some fundamental estimates for solutions to \eqref{I1}
 and prove Theorem \ref{thm1}. In Section 4, we consider the large
time behavior of solutions to \eqref{I1} for a special case and prove
Theorem \ref{thm2}.

\section{Preliminaries}

 In this section, we first state the local existence of solutions to the
system \eqref{I1} and then present some preliminary inequalities.
 By directly adapting the reasoning in \cite [Lemma 2.1]{T-W-SIAM},
we can derive the following lemma on local existence of classical
solutions to \eqref{I1}.

\begin{lemma}\label{lem21}
Let $\Omega\subset \mathbb{R}^{3}$ be a bounded domain with smooth boundary
and initial data $(u_0,v_0,w_0)$ satisfy \eqref{I2}.
Suppose that $m\geq1$, $r\geq 0$, $\mu>0$, $\eta>1$, $\chi>0$, $\xi>0$,
 $\alpha>0$, $\beta>0$, $\gamma >0$ and $\delta>0$.
Then there exist $ T_{\rm max}\in(0,\infty]$ and a triplet
$(u,v,w)$ of nonnegative functions from
$C^{0}\big(\bar{\Omega}\times[0,T_{\rm max})\big)\cap C^{2,1}\big(\bar{\Omega}
\times (0, T_{\rm max})\big)$ solving \eqref{I1} classically in
$\Omega\times(0,T_{\rm max})$. These functions satisfy $u>0$, $v>0$ and
$w>0$ in $\bar{\Omega} \times (0, T_{\rm max})$ and, moreover, we have
\begin{equation}\label{Pre1}
\begin{aligned}
&\text{either $T_{\rm max} = \infty$, or}\\
&\| u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}
+ \|w(\cdot,t)\|_{W^{1,\infty}(\Omega)}\to\infty\text{ as }t\to T_{\rm max}.
\end{aligned}
\end{equation}
\end{lemma}

Let us state the following basic properties associated with $u$.

\begin{lemma}\label{lem22}
There exists some constants $M_1>0$ and $C>0$ such that
 \begin{gather}\label{Pre2}
 \int_{\Omega} u(x,t)dx\leq M_1 \quad \forall t\in(0,T_{\rm max}), \\
\label{Pre3}
 \int^{t+\tau}_t\int_{\Omega} u^\eta (x,t)\,dx\,ds\leq C \quad \forall  t\in(0,T_{\rm max}-\tau),
 \end{gather}
 where
 \begin{equation}\label{Pre4}
 \tau:=\min\{1,\frac{1}{2}T_{\rm max}\}.
 \end{equation}
\end{lemma}

\begin{proof}
Integrating the first equation of \eqref{I1}, we obtain
\begin{equation}\label{Pre5}
 \frac{d}{dt}\int_\Omega u dx+\mu\int_\Omega u^\eta dx =r\int_\Omega udx
 \quad\forall  t\in(0,T_{\rm max}).
\end{equation}
If $r=0$, we have $\frac{d}{dt}\int_\Omega u dx=-\mu\int_\Omega u^\eta dx$
for all $t\in(0,T_{\rm max})$. The nonnegativity of $u$ implies
$\frac{d}{dt}\int_\Omega u dx<0$ for all $t\in(0,T_{\rm max})$. Thus, we have
$\int_\Omega u dx \leq \int_\Omega u_0 dx$ for all $t\in(0,T_{\rm max})$.
If $r>0$, then by using Young's inequality we have
\[
2ru \leq \mu u^\eta+\mu^{-\frac{1}{\eta-1}}(2r)^{\frac{\eta}{\eta-1}}.
\]
This implies
\[
\frac{d}{dt}\int_\Omega u dx+r\int_\Omega u dx
\leq \mu^{-\frac{1}{\eta-1}}(2r)^{\frac{\eta}{\eta-1}}|\Omega|
 \quad\forall t\in(0,T_{\rm max}).
\]
Thus,  
\[
\int_\Omega udx\leq M_1:=\max\Big\{\int_\Omega u_0dx,
\frac{ \mu^{-\frac{1}{\eta-1}}(2r)^{\frac{\eta}{\eta-1}}|\Omega|}{r}\Big\}
\]
for all $t\in(0,T_{\rm max})$. Therefore, \eqref{Pre2} holds.
By a time integration of \eqref{Pre5}, we derive that
 \[
 \begin{split}
 &\int_\Omega u(x,t+\tau)dx +\mu \int^{t+\tau}_t\int_\Omega u^\eta(x,s) \,dx\,ds\\
 & \leq \int_\Omega u(x,t)dx +r\int^{t+\tau}_t\int_\Omega u(x,s) \,dx\,ds \\
 & \leq M_1+r\tau M_1
 \quad\forall  t\in(0,T_{\rm max}-\tau),
 \end{split}
 \]
 which implies that \eqref{Pre3} holds.
\end{proof}

The following auxiliary lemma on a boundedness property in an ODI will
 be used in our analysis. It is a straightforward generalization of a
particular case $\tau=1$ which has been proved in \cite[Lemma 3.4]{SSW-SIAM}.

 \begin{lemma}\label{lem23}
Let $T>0$, $\tau\in(0,T)$, $a>0$ and $b>0$, and suppose that
$y: [0,T)\to [0,\infty)$ is absolutely continuous and such that
\[
y'(t)+ay(t)\leq h(t) \quad \text{for a.e. } t\in(0,T),
\]
where $h\in L^1_{loc}([0,T))$ is nonnegative and satisfies
\[
\int^{t+\tau}_t h(s)ds\leq b \quad \forall  t\in[0, T-\tau).
\]
 Then
\[
y(t)\leq \max\{y(0)+b, \frac{b}{a\tau}+2b\} \quad \text{for all } t\in(0,T).
\]
\end{lemma}

As an application of Lemma \ref{lem22} and Lemma \ref{lem23},
we can establish the following estimates associated with $v$ and $w$.

\begin{lemma}\label{lem24}
Let $\eta\geq 2$, then there exists a constant $C>0$ such that
\begin{gather}\label{Pre6}
 \int_\Omega v(x,t)dx \leq C \quad \forall t\in(0, T_{\rm max}),\\
\label{Pre7}
 \int_\Omega |\nabla v|^2 dx \leq C \quad \forall t\in(0, T_{\rm max}),\\
\label{Pre8}
 \int_\Omega w(x,t)dx \leq C \quad \forall t\in(0, T_{\rm max}), \\
\label{Pre9}
 \int_\Omega |\nabla w|^2 dx \leq C \quad \forall t\in(0, T_{\rm max}).
\end{gather}
\end{lemma}

\begin{proof}
We first prove the estimates associated with $v$.
Integrating the second equation in \eqref{I1} over $\Omega$ and then
using \eqref{Pre2}, we obtain that
\[
\frac{d}{dt}\int_\Omega vdx = -\beta \int_\Omega vdx +\alpha \int_\Omega udx
\leq -\beta \int_\Omega vdx +\alpha M_1 \quad \forall t\in(0, T_{\rm max}),
\]
which implies \eqref{Pre6} by an ODE comparison. We multiply the second equation in \eqref{I1} by $-\Delta v$ and integrate by parts over $\Omega$ to derive that
\begin{equation}\label{Pre10}
 \begin{split}
&\frac{1}{2}\frac{d}{dt}\int_\Omega |\nabla v|^2dx
+ \int_\Omega |\Delta v|^2dx +\beta \int_\Omega |\nabla v|^2dx \\
&=-\alpha \int_\Omega u \Delta v\\
&\leq \frac{1}{2}\int_\Omega |\Delta v|^2dx+ \frac{\alpha^2}{2}\int_\Omega u^2dx
 \quad \forall t\in(0, T_{\rm max}).
 \end{split}
\end{equation}
Letting $y(t):=\int_\Omega |\nabla v|^2dx$ and
$h(t):=\alpha^2\int_\Omega u^2dx$ for $t\in(0,T_{\rm max})$, we have
\[
y'(t)+2\beta y(t)\leq h(t) \quad \forall t\in(0, T_{\rm max}).
\]
Since \eqref{Pre3} holds for all $\eta\geq 2$, we have
$\int^{t+\tau}_t h(s)ds\leq C_1$ for any $t\in (0, T_{\rm max}-\tau)$
with some constant $C_1>0$ and $\tau=\min\left\{1, \frac{1}{2}T_{\rm max}\right\}$.
Thus, according to Lemma \ref{lem23}, we establish \eqref{Pre7}.
Taking a similar procedure for $w$, we can also establish \eqref{Pre8}
and \eqref{Pre9}.
\end{proof}

\section{Global existence and boundedness of classical solutions }

 In this section, we investigate the global existence and boundedness
of classical solutions to system \eqref{I1}.
In Section 3.1, we establish the global existence for the strong logistic
source case (i.e., $\eta\geq 2$ and
$\mu>\max\big\{1+4\chi^2+11\alpha^2+2\chi^2(9+\alpha^2+\gamma^2)$,
$1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2) \big\}$).
In Section 3.2, we establish it for the sufficiently strong diffusion
 case (i.e., $m>\frac{4}{3}$). The key idea is to deduce a uniform estimate for
 $\| u(\cdot,t)\|_{L^p(\Omega)}$+$\|\nabla v(\cdot,t)\|_{L^q(\Omega)}$+
 $\|\nabla w(\cdot,t)\|_{L^q(\Omega)}$ for sufficiently large $p$ and $q$.

 \subsection*{A priori estimates for the strong logistic source case}
Inspired by Tao and Winkler \cite{T-W-ZAMP}, we first establish an
estimate for an appropriate linear combination of the functions
$\int_\Omega (u+1)^{m+1}dx$, $\int_\Omega |\nabla v|^4dx$,
$\int_\Omega |\nabla w|^4dx$, $\int_\Omega u|\nabla v|^2dx$ and
$\int_\Omega u|\nabla w|^2dx$. As a beginning, let us derive
differential inequalities for $\int_\Omega (u+1)^{m+1}dx$,
$\int_\Omega |\nabla v|^4dx$ and $\int_\Omega |\nabla w|^4dx$.

\begin{lemma}\label{lem31}
 Let $\eta\geq 2$ and the other assumptions in Lemma \ref{lem21} hold.
Then we have
 \begin{equation}\label{Prf3-1}
 \begin{split}
&\frac{d}{dt}\int_\Omega (u+1)^{m+1}dx
 + \frac{m(m+1)}{2}\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx\\
&\leq \chi^2m(m+1)\int_\Omega u^2|\nabla v|^2dx +\xi^2m(m+1)
 \int_\Omega u^2|\nabla w|^2dx \\
&\quad +r(m+1)\int_\Omega (u+1)^{m+1}dx-\frac{\mu(m+1)}{2^\eta}
 \int_\Omega (u+1)^{m+\eta}dx \\
&\quad +\mu(m+1)\int_\Omega (u+1)^mdx \quad \forall t\in(0,T_{\rm max}).
 \end{split}
\end{equation}
\end{lemma}

\begin{proof}
Multiplying the first equation in \eqref{I1} by $(u+1)^m$ and
integrating by parts over $\Omega$, we obtain
\begin{align*}
&\frac{1}{m+1}\frac{d}{dt}\int_\Omega (u+1)^{m+1}dx \\
&=  -m\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx
 +\chi m \int_\Omega u(u+1)^{m-1}\nabla v\cdot \nabla u dx\\
&\quad-\xi m \int_\Omega u(u+1)^{m-1}\nabla w\cdot \nabla u dx
 +r\int_\Omega u(u+1)^mdx-\mu\int_\Omega u^\eta(u+1)^mdx \\
&\leq -\frac{m}{2}\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx
 +\chi^2 m\int_\Omega u^{2}|\nabla v|^2 dx\\
&\quad +\xi^2m\int_\Omega u^{2}|\nabla w|^2dx
 +r\int_\Omega (u+1)^{m+1}dx-\mu\int_\Omega u^\eta(u+1)^mdx
\end{align*}
for all $t\in(0,T_{\rm max})$. Due to the nonnegativity of $u$,
we have $(u+1)^\eta\leq 2^\eta(u^\eta+1)$ and then obtain
$u^\eta\geq \frac{1}{2^\eta}(u+1)^\eta-1$.
Inserting it into the above inequality, we obtain
the desired result \eqref{Prf3-1}.
\end{proof}

\begin{lemma}\label{lem32}
Let the assumptions in Lemma \ref{lem31} hold. Then we have
\begin{gather}\label{Prf3-2}
\begin{aligned}
&\frac{d}{dt}\int_\Omega |\nabla v|^4dx
 +\frac{3}{2}\int_\Omega |\nabla |\nabla v|^2|^2dx \\
&\leq 2\int_{\partial\Omega}|\nabla v|^2
 \frac{\partial |\nabla v|^2}{\partial\nu}dx
 + 11\alpha^2\int_\Omega u^2|\nabla v|^2dx, \quad\forall  t\in(0, T_{\rm max}),
\end{aligned} \\
\label{Prf3-3}
\begin{aligned}
&\frac{d}{dt}\int_\Omega |\nabla w|^4dx+\frac{3}{2}\int_\Omega
|\nabla |\nabla w|^2|^2dx \\
&\leq 2\int_{\partial\Omega}|\nabla w|^2
 \frac{\partial |\nabla w|^2}{\partial\nu}dx
 + 11\gamma^2\int_\Omega u^2|\nabla w|^2dx, \quad\forall  t\in(0, T_{\rm max})\,.
\end{aligned}
\end{gather}
\end{lemma}

\begin{proof}
 Differentiating the second equation in \eqref{I1} and then multiplying
$2\nabla v$, we obtain that
\begin{equation}\label{Prf3-4}
 (|\nabla v|^2)_t=\Delta |\nabla v|^2-2|D^2v|^2
-2\beta|\nabla v|^2+2\alpha\nabla u\cdot \nabla v \quad
\forall t\in(0, T_{\rm max}),
\end{equation}
where we have used the identity
$\Delta |\nabla v|^2=2\nabla v\cdot \nabla\Delta v+2|D^2v|^2$.
Testing \eqref{Prf3-4} against $2|\nabla v|^2$ yields
\begin{equation}\label{Prf3-5}
\begin{split}
\frac{d}{dt}\int_\Omega |\nabla v|^4dx
& = 2\int_\Omega \Delta |\nabla v|^2 |\nabla v|^2dx
 -4\int_\Omega |D^2 v|^2|\nabla v|^2dx\\
&\quad -4\beta \int_\Omega|\nabla v|^4dx
 +4\alpha \int_\Omega (\nabla u\cdot\nabla v)|\nabla v|^2dx\\
 &=2 \int_{\partial\Omega}|\nabla v|^2\frac{\partial|\nabla v|^2}{\partial\nu}dx
 -2\int_\Omega |\nabla |\nabla v|^2|^2dx \\
 &\quad -4\int_\Omega |D^2 v|^2|\nabla v|^2dx-4\beta \int_\Omega|\nabla v|^4dx\\
 &\quad-4\alpha \int_\Omega u\Delta v |\nabla v|^2dx
-4\alpha \int_\Omega u\nabla v\cdot\nabla|\nabla v|^2dx
 \end{split}
\end{equation}
for all $t\in(0, T_{\rm max})$.
Since $|\Delta v|\leq \sqrt{3}|D^2v|$, by Young's inequality we have
\begin{equation}\label{Prf3-6}
\begin{split}
 -4\alpha \int_\Omega u\Delta v |\nabla v|^2dx
 \leq & 4\sqrt{3}\alpha \int_\Omega u|D^2v| |\nabla v|^2dx\\
 \leq & 4 \int_\Omega |D^2 v|^2|\nabla v|^2dx +3\alpha^2 \int_\Omega u^2|\nabla v|^2dx
 \end{split}
\end{equation}
and
\begin{equation}\label{Prf3-7}
 -4\alpha \int_\Omega u\nabla v\cdot\nabla|\nabla v|^2dx
 \leq \frac{1}{2}\int_\Omega |\nabla |\nabla v|^2|^2dx
 +8\alpha^2\int_\Omega u^2|\nabla v|^2dx
\end{equation}
for all $t\in(0, T_{\rm max})$. Inserting \eqref{Prf3-6} and
\eqref{Prf3-7} into \eqref{Prf3-5}, we infer that
\begin{equation}\label{Prf3-8}
\begin{split}
\frac{d}{dt}\int_\Omega |\nabla v|^4dx
&\leq  2 \int_{\partial\Omega}|\nabla v|^2\frac{\partial|\nabla v|^2}{\partial\nu}dx
 -\frac{3}{2}\int_\Omega |\nabla |\nabla v|^2|^2dx \\
&\quad +11\alpha^2\int_\Omega u^2|\nabla v|^2dx  -4\beta \int_\Omega|\nabla v|^4dx\\
&\leq 2 \int_{\partial\Omega}|\nabla v|^2
 \frac{\partial|\nabla v|^2}{\partial\nu}dx
 -\frac{3}{2}\int_\Omega |\nabla |\nabla v|^2|^2dx\\
&\quad +11\alpha^2 \int_\Omega u^2|\nabla v|^2dx
 \quad \forall t\in(0, T_{\rm max}).
 \end{split}
\end{equation}
Thus, we obtain \eqref{Prf3-2}. Taking a same procedure for the third
 equation in \eqref{I1}, we can derive \eqref{Prf3-3}. \
\end{proof}

To deal with the integrals $\int_\Omega u^2|\nabla v|^2dx$ and
$\int_\Omega u^2|\nabla w|^2dx$ on the right-hand side of
\eqref{Prf3-1}-\eqref{Prf3-3}, we shall establish the following
 differential inequality related to them.

\begin{lemma}\label{lem33}
Let the assumptions in Lemma \ref{lem31} hold. Then we have
 \begin{align}
 &\frac{d}{dt}\Big\{\int_\Omega u|\nabla v|^2dx+\int_\Omega u|\nabla w|^2dx\Big\}
\nonumber \\
 &+(\mu-1-4\chi^2)\int_\Omega u^2|\nabla v|^2dx+(\mu-1-4\xi^2)
 \int_\Omega u^2|\nabla w|^2dx \nonumber\\
& \leq\frac{1}{2}\int_\Omega |\nabla |\nabla v|^2|^2dx
 +\frac{1}{2}\int_\Omega |\nabla |\nabla w|^2|^2dx \nonumber\\
&\quad +(8+\alpha^2+\gamma^2)\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx
 +(r-2\beta)\int_\Omega u|\nabla v|^2dx \nonumber\\
&\quad +(r-2\delta)\int_\Omega u|\nabla w|^2dx
 +\int_{\partial\Omega}u\frac{\partial|\nabla v|^2}{\partial\nu}dx
 +\int_{\partial\Omega}u\frac{\partial|\nabla w|^2}{\partial\nu}dx+C
\label{Prf3-9}
 \end{align}
for all $t\in(0,T_{\rm max})$, where $C$ is a positive constant.
\end{lemma}

\begin{proof}
 By employing the identity
$\nabla v\cdot \nabla \Delta v=\frac{1}{2}\Delta |\nabla v|^2-|D^2v|^2$,
we take a straightforward computation on
$\frac{d}{dt}\int_\Omega u|\nabla v|^2dx$ to obtain that
\begin{equation}\label{Prf3-10}
\begin{split}
&\frac{d}{dt}\int_\Omega u|\nabla v|^2dx \\
&=  \int_\Omega |\nabla v|^2\left\{\nabla\cdot((u+1)^{m-1}
 \nabla u-\chi u\nabla v+\xi u\nabla w)+ru-\mu u^\eta\right\}dx \\
&\quad+2\int_\Omega u\nabla v\cdot\nabla(\Delta v-\beta v+\alpha u)dx\\
&= -\int_\Omega (u+1)^{m-1}\nabla u\cdot \nabla|\nabla v|^2dx
 +\chi \int_\Omega u\nabla v\cdot \nabla|\nabla v|^2dx \\
&\quad -\xi\int_\Omega u\nabla w\cdot \nabla|\nabla v|^2dx
 -\mu\int_\Omega u^\eta|\nabla v|^2dx\\
&\quad +2\int_\Omega u(\frac{1}{2}\Delta |\nabla v|^2-|D^2v|^2)dx+
 (r-2\beta)\int_\Omega u|\nabla v|^2dx\\
&\quad +2\alpha\int_\Omega u\nabla v\cdot \nabla udx
\end{split}
\end{equation}
for all $t\in(0, T_{\rm max})$. We now estimate the integrals on the right
 of \eqref{Prf3-10}. By Young's inequality, we have
\begin{gather}\label{Prf3-11}
\begin{aligned}
&-\int_\Omega (u+1)^{m-1}\nabla u\cdot \nabla|\nabla v|^2dx\\
&\leq \frac{1}{8}\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
+2\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx,
\end{aligned} \\
\label{Prf3-12}
 \chi \int_\Omega u\nabla v\cdot \nabla|\nabla v|^2dx
\leq \frac{1}{8}\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
+2\chi^2\int_\Omega u^2|\nabla v|^2dx, \\
\label{Prf3-13}
 -\xi\int_\Omega u\nabla w\cdot \nabla|\nabla v|^2dx
\leq \frac{1}{8}\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
+2\xi^2\int_\Omega u^2|\nabla w|^2dx, \\
\label{Prf3-14}
 -\mu\int_\Omega u^\eta|\nabla v|^2dx
\leq \mu\int_\Omega |\nabla v|^2dx -\mu \int_\Omega u^2 |\nabla v|^2dx, \\
\label{Prf3-15}
\begin{split}
&2\int_\Omega u(\frac{1}{2}\Delta |\nabla v|^2-|D^2v|^2)dx \\
 &= \int_\Omega u \Delta |\nabla v|^2dx-2\int_\Omega u |D^2v|^2dx \\
 &=  \int_{\partial\Omega} u\frac{\partial |\nabla v|^2}{\partial\nu}dx-
 \int_\Omega \nabla |\nabla v|^2\cdot \nabla udx -2\int_\Omega u |D^2v|^2dx \\
&\leq \int_{\partial\Omega} u\frac{\partial |\nabla v|^2}{\partial\nu}dx
 +\frac{1}{8}\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
+2\int_\Omega |\nabla u|^2dx,
 \end{split} \\
\label{Prf3-16}
 2\alpha\int_\Omega u\nabla v\cdot \nabla udx
\leq \int_\Omega u^2|\nabla v|^2dx+\alpha^2\int_\Omega |\nabla u|^2dx
\end{gather}
for all $t\in(0,T_{\rm max})$. Substituting \eqref{Prf3-11}-\eqref{Prf3-16}
into \eqref{Prf3-10} and using \eqref{Pre7}, we obtain that there exists
a positive constant $C_1$ satisfying
\begin{equation}\label{Prf3-18}
 \begin{split}
&\frac{d}{dt}\int_\Omega u |\nabla v|^2dx
 +(\mu-1-2\chi^2)\int_\Omega u^2 |\nabla v|^2dx\\
&\leq \frac{1}{2}\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
+(4+\alpha^2)\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx\\
&\quad +2\xi^2\int_\Omega u^2|\nabla w|^2dx+(r-2\beta)\int_\Omega u|\nabla v|^2dx
+\int_{\partial\Omega} u\frac{\partial |\nabla v|^2}{\partial\nu}dx
+C_1
\end{split}
\end{equation}
for all $t\in(0,T_{\rm max})$. By taking a similar procedure, we can derive that
\begin{equation}\label{Prf3-19}
 \begin{split}
&\frac{d}{dt}\int_\Omega u |\nabla w|^2dx
 +(\mu-1-2\xi^2)\int_\Omega u^2 |\nabla w|^2dx\\
&\leq \frac{1}{2}\int_\Omega \big|\nabla|\nabla w|^2\big|^2dx
+(4+\gamma^2)\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx\\
&\quad +2\chi^2\int_\Omega u^2|\nabla v|^2dx
 +(r-2\delta)\int_\Omega u|\nabla w|^2dx
+\int_{\partial\Omega} u\frac{\partial |\nabla w|^2}{\partial\nu}dx +C_2
\end{split}
\end{equation}
with a constant $C_2>0$ for all $t\in(0, T_{\rm max})$.
Adding \eqref{Prf3-19} to \eqref{Prf3-18} yields \eqref{Prf3-9}.
\end{proof}

Multiplying \eqref{Prf3-1} by $\frac{2(9+\alpha^2+\gamma^2)}{m(m+1)}$
and then combining \eqref{Prf3-2}, \eqref{Prf3-3} and \eqref{Prf3-9},
 we obtain the following result.

\begin{corollary}\label {Coro31}
Let the assumptions in Lemma \ref{lem31} hold. Then we have
\begin{equation}\label{Prf3-20}
\begin{split}
 & \frac{d}{dt}\Big\{\frac{2(9+\alpha^2+\gamma^2)}{m(m+1)}\int_\Omega (u+1)^{m+1}dx
 +\int_\Omega |\nabla v|^4dx+\int_\Omega |\nabla w|^4dx \\
 & +\int_\Omega u|\nabla v|^2dx+\int_\Omega u|\nabla w|^2dx \Big\}
 +\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
 +\int_\Omega \big|\nabla|\nabla w|^2\big|^2dx \\
 & +[\mu-1-4\chi^2-11\alpha^2-2\chi^2(9+\alpha^2+\gamma^2)]
 \int_\Omega u^2|\nabla v|^2dx\\
 &+[\mu-1-4\xi^2-11\gamma^2-2\xi^2(9+\alpha^2+\gamma^2)]\int_\Omega u^2|\nabla w|^2dx\\
 &+\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx
 +\frac{2\mu(9+\alpha^2+\gamma^2)}{2^\eta m}\int_\Omega (u+1)^{m+\eta}dx\\
&\leq  (r-2\beta)\int_\Omega u|\nabla v|^2dx +(r-2\delta)\int_\Omega u|\nabla w|^2dx\\
&\quad + \frac{2(9+\alpha^2+\gamma^2)}{m}\Big(r\int_\Omega (u+1)^{m+1}dx
 +\mu\int_\Omega (u+1)^mdx\Big)\\
 &\quad +\int_{\partial\Omega} u\frac{\partial |\nabla v|^2}{\partial\nu}dx
 +\int_{\partial\Omega} u\frac{\partial |\nabla w|^2}{\partial\nu}dx
 +2\int_{\partial\Omega} |\nabla v|^2\frac{\partial |\nabla v|^2}{\partial\nu}dx \\
&\quad +2\int_{\partial\Omega} |\nabla w|^2\frac{\partial |\nabla w|^2}{\partial\nu}dx+C
 \end{split}
\end{equation}
for all $t\in(0,T_{\rm max})$, where $C$ is a positive constant.
\end{corollary}

We now are in a position to establish the estimates on
$\int_\Omega u^{m+1}dx$, $\int_\Omega |\nabla v|^4dx$ and
$\int_\Omega |\nabla w|^4dx$. We will show that if $\mu$ is taken large enough,
then these integrals are uniformly bounded for all $t\in(0,T_{max})$.

\begin{lemma}\label{lem34}
 Suppose that $\mu>\max\{1+4\chi^2+11\alpha^2+2\chi^2(9+\alpha^2+\gamma^2),
1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2) \}$.
Let $\eta \geq 2$ and let $(u,v,w)$ be a solution to system \eqref{I1}.
Then we can find a constant $C>0$ such that
 \begin{gather}\label{Prf3-21}
 \int_\Omega (u+1)^{m+1}dx \leq C \quad \forall t\in(0, T_{\rm max}),\\
\label{Prf3-22}
 \int_\Omega |\nabla v|^4dx \leq C \quad \forall t\in(0, T_{\rm max}), \\
\label{Prf3-23}
 \int_\Omega |\nabla w|^4dx \leq C \quad \forall t\in(0, T_{\rm max}).
 \end{gather}
\end{lemma}

\begin{proof}
Let $y(t):=\frac{2(9+\alpha^2+\gamma^2)}{m(m+1)}\int_\Omega (u+1)^{m+1}dx
 +\int_\Omega |\nabla v|^4dx+\int_\Omega |\nabla w|^4dx+ \\
\int_\Omega u|\nabla v|^2dx+\int_\Omega u|\nabla w|^2dx$.
Since $\mu>\max\{1+4\chi^2+11\alpha^2+2\chi^2(9+\alpha^2+\gamma^2),
1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2)\}$,
from Corollary \ref{Coro31} we have
 \begin{equation}\label{Prf3-24}
 \begin{split}
 & y'(t)+y(t)+\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
 +\int_\Omega \big|\nabla|\nabla w|^2\big|^2dx\\
&+\int_\Omega (u+1)^{2(m-1)} |\nabla u|^2dx
 +\frac{2\mu(9+\alpha^2+\gamma^2)}{2^\eta m}\int_\Omega (u+1)^{m+\eta}dx\\
 &\leq (r+1-2\beta)\int_\Omega u|\nabla v|^2dx +(r+1-2\delta)
 \int_\Omega u|\nabla w|^2dx \\
 &\quad + \frac{2(9+\alpha^2+\gamma^2)[r(m+1)+1]}{m(m+1)}\int_\Omega (u+1)^{m+1}dx\\
 &\quad +\int_{\partial\Omega} u\frac{\partial |\nabla v|^2}{\partial\nu}dx
 +\int_{\partial\Omega} u\frac{\partial |\nabla w|^2}{\partial\nu}dx
 +2\int_{\partial\Omega} |\nabla v|^2\frac{\partial |\nabla v|^2}{\partial\nu}dx\\
&\quad +2\int_{\partial\Omega} |\nabla w|^2\frac{\partial
 |\nabla w|^2}{\partial\nu}dx
  +\frac{2\mu(9+\alpha^2+\gamma^2)}{m}\int_\Omega (u+1)^mdx\\
 &\quad +\int_\Omega |\nabla v|^4dx+\int_\Omega |\nabla w|^4dx+C_0
 \quad \forall t\in(0, T_{\rm max}),
 \end{split}
 \end{equation}
 where $C_0$ is a positive constant.
 We next use the dissipated quantities on the left of \eqref{Prf3-24}
to estimate all integrals on the right. By the Gagliardo-Nirenberg inequality,
we obtain that
 \begin{equation}\label{Prf3-25}
\begin{aligned}
\int_\Omega |\nabla v|^4dx
&=\||\nabla v|^2\|^2_{L^2(\Omega)}\\
&\leq C_1\|\nabla|\nabla v|^2\|^{\frac{6}{5}}_{L^2(\Omega)}
 \||\nabla v|^2\|^{\frac{4}{5}}_{L^1(\Omega)}+C_1\||\nabla v|^2\|^2_{L^1(\Omega)}
\end{aligned}
 \end{equation}
for all $t\in(0, T_{\rm max})$, with some constant $C_1>0$.
From Lemma \ref{lem24}, we know that
$\int_\Omega |\nabla v|^2\leq C_2$ for all $t\in(0,T_{\rm max})$
 with some $C_2>0$. Thus by Young's inequality we can find a constant
$C_3>0$ such that
\begin{equation}\label{Prf3-26}
 \int_\Omega |\nabla v|^4dx\leq \frac{1}{8}\|\nabla|\nabla v|^2\|^2_{L^2(\Omega)}+C_3
 \quad \forall t\in(0, T_{\rm max}).
\end{equation}
Similarly, we can also have
\begin{equation}\label{Prf3-27}
\int_\Omega |\nabla w|^4dx\leq \frac{1}{8}\|\nabla|\nabla w|^2\|^2_{L^2(\Omega)}+C_4
 \quad \forall t\in(0, T_{\rm max})
\end{equation}
with some constant $C_4>0$. By Young's inequality, we obtain
\begin{equation}\label{Prf3-28}
 (r+1-2\beta)\int_\Omega u|\nabla v|^2dx
 \leq \frac{(r+1-2\beta)^2}{4}\int_\Omega u^2dx+\int_\Omega |\nabla v|^4dx
\end{equation}
and hence, upon \eqref{Prf3-26} we derive that
\begin{equation}\label{Prf3-29}
\begin{split}
&(r+1-2\beta)\int_\Omega u|\nabla v|^2dx+\int_\Omega |\nabla v|^4dx \\
 &\leq  \frac{(r+1-2\beta)^2}{4}\int_\Omega u^2dx+2\int_\Omega |\nabla v|^4dx\\
 &\leq  \frac{(r+1-2\beta)^2}{4}\int_\Omega u^2dx
 +\frac{1}{4}\|\nabla|\nabla v|^2\|^2_{L^2(\Omega)}+2C_3\\
 &\leq  \frac{\mu(9+\alpha^2+\gamma^2)}{2^{\eta+1} m}
 \int_\Omega (u+1)^{m+\eta}dx+\frac{1}{4}\|\nabla|\nabla v|^2\|^2_{L^2(\Omega)}+C_5
 \end{split}
\end{equation}
with some constant $C_5>0$ for all $t\in(0,T_{\rm max})$.
Similarly, we can obtain
\begin{equation}\label{Prf3-30}
\begin{split}
&(r+1-2\delta)\int_\Omega u|\nabla w|^2dx+\int_\Omega |\nabla w|^4dx\\
 &\leq  \frac{\mu(9+\alpha^2+\gamma^2)}{2^{\eta+1} m}\int_\Omega (u+1)^{m+\eta}dx+\frac{1}{4}\|\nabla|\nabla w|^2\|^2_{L^2(\Omega)}+C_6
 \end{split}
\end{equation}
with some $C_6>0$ for all $t\in(0,T_{\rm max})$.
Using Young's inequality again, we obtain that there exist positive
constants $C_7$ and $C_8$ fulfilling
\begin{gather}\label{Prf3-31}
\begin{aligned}
&\frac{2(9+\alpha^2+\gamma^2)[r(m+1)+1]}{m(m+1)}\int_\Omega (u+1)^{m+1}dx \\
&\leq \frac{\mu(9+\alpha^2+\gamma^2)}{2^{\eta+1} m}
\int_\Omega (u+1)^{m+\eta}dx + C_7,
\end{aligned} \\
\label{Prf3-32}
\begin{aligned}
&\frac{2\mu(9+\alpha^2+\gamma^2)}{m}\int_\Omega (u+1)^mdx \\
&\leq \frac{\mu(9+\alpha^2+\gamma^2)}{2^{\eta+1} m}
\int_\Omega (u+1)^{m+\eta}dx + C_8
\end{aligned}
\end{gather}
for all $t\in(0,T_{\rm max})$. For the boundary integrals in \eqref{Prf3-24},
by taking similar arguments in \cite[(3.20)]{T-W-ZAMP} we obtain that there
exist some positive constants $C_9$ and $C_{10}$ fulfilling
\begin{gather}\label{Prf3-33}
\begin{aligned}
&\int_{\partial\Omega} u\frac{\partial |\nabla v|^2}{\partial\nu}dx
 +2\int_{\partial\Omega} |\nabla v|^2\frac{\partial |\nabla v|^2}{\partial\nu}dx\\
&\leq \frac{1}{4}\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
 + \frac{1}{2}\int_\Omega |\nabla u|^2dx + C_9 ,
\end{aligned}\\
\label{Prf3-34}
\begin{aligned}
&\int_{\partial\Omega} u\frac{\partial |\nabla w|^2}{\partial\nu}dx
 +2\int_{\partial\Omega} |\nabla w|^2\frac{\partial |\nabla w|^2}{\partial\nu}dx\\
&\leq \frac{1}{4}\int_\Omega \big|\nabla|\nabla w|^2\big|^2dx
 + \frac{1}{2}\int_\Omega |\nabla u|^2dx + C_{10}
\end{aligned}
\end{gather}
for all $t\in(0,T_{\rm max})$. Substituting \eqref{Prf3-29}-\eqref{Prf3-34}
into \eqref{Prf3-24}, we obtain that
\begin{equation}\label{Prf3-341}
 \begin{split}
 & y'(t)+y(t)+\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
 +\int_\Omega \big|\nabla|\nabla w|^2\big|^2dx\\
 & +\int_\Omega (u+1)^{2(m-1)}|\nabla u|^2dx
 +\frac{2\mu(9+\alpha^2+\gamma^2)}{2^\eta m}\int_\Omega (u+1)^{m+\eta}dx\\
 &\leq \frac{1}{2}\int_\Omega \big|\nabla|\nabla v|^2\big|^2dx
 +\frac{1}{2}\int_\Omega \big|\nabla|\nabla w|^2\big|^2dx \\
&\quad +\int_\Omega |\nabla u|^2dx+\frac{2\mu(9+\alpha^2+\gamma^2)}{2^\eta m}
\int_\Omega (u+1)^{m+\eta}dx+C_{11}
 \end{split}
\end{equation}
for all $t\in(0,T_{\rm max})$, where $C_{11}:=C_0+\sum^{10}_{i=5}C_i$.
Hence we have
\begin{equation}\label{Prf3-342}
 y'(t)+y(t)\leq C_{11} \quad \forall t\in(0, T_{\rm max}).
\end{equation}
Thus, by Gronwall's inequality, we have
\begin{align*}
 y(t)
&=\frac{2(9+\alpha^2+\gamma^2)}{m(m+1)}\int_\Omega u^{m+1}dx
  +\int_\Omega |\nabla v|^4dx+\int_\Omega |\nabla w|^4dx \\
&\quad +\int_\Omega u|\nabla v|^2dx+\int_\Omega u|\nabla w|^2dx \\
&\leq \max\{y(0), C_{11}\}
\end{align*}
for all $t\in(0, T_{\rm max})$, which implies \eqref{Prf3-21}, \eqref{Prf3-22}
and \eqref{Prf3-23}.
\end{proof}

We now use the $L^{m+1}$ estimate for $u+1$ and the $L^4$ estimate for
$\nabla v$ and $\nabla w$ from Lemma \ref{lem34} to establish higher
regularity estimates for $(u,v,w)$.

\begin{lemma}\label{lem35}
Suppose that $\mu>\max\{1+4\chi^2+11\alpha^2
+2\chi^2(9+\alpha^2+\gamma^2), 1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2)
\}$. Let $\eta \geq 2$ and let $(u,v,w)$ be a solution to system \eqref{I1}.
 Then for all $p>\frac{3m-1}{2}$ and each $q>1$ there exists a constant
$C>0$ such that
\begin{gather}\label{Prf3-35}
 \|u(\cdot, t)\|_{L^p(\Omega)}\leq C \quad \forall t\in(0,T_{\rm max}),\\
\label{Prf3-36}
 \|\nabla v(\cdot, t)\|_{L^q(\Omega)}\leq C \quad \forall t\in(0,T_{\rm max}),\\
\label{Prf3-37}
 \|\nabla w(\cdot, t)\|_{L^q(\Omega)}\leq C \quad \forall t\in(0,T_{\rm max}).
\end{gather}
\end{lemma}

\begin{proof}
Testing the first equation in \eqref{I1} against $(u+1)^{p-1}$ and then
integrating by parts over $\Omega$, we derive that
 \begin{align}
&\frac{1}{p}\frac{d}{dt}\int_\Omega (u+1)^pdx \nonumber \\
&= -(p-1)\int_\Omega(u+1)^{m+p-3}|\nabla u|^2dx
+(p-1)\chi \int_\Omega u(u+1)^{p-2}\nabla v\cdot \nabla udx \nonumber\\
 &\quad -(p-1)\xi \int_\Omega u(u+1)^{p-2}\nabla w\cdot \nabla udx+
 r\int_\Omega u(u+1)^{p-1}dx \nonumber\\
&\quad - \mu \int_\Omega u^\eta (u+1)^{p-1}dx \nonumber\\
 &\leq -(p-1)\int_\Omega(u+1)^{m+p-3}|\nabla u|^2dx
+(p-1)\chi \int_\Omega (u+1)^{p-1}|\nabla v| |\nabla u|dx \nonumber\\
 &\quad +(p-1)\xi \int_\Omega (u+1)^{p-1}|\nabla w||\nabla u|dx
 +r\int_\Omega (u+1)^{p}dx \nonumber\\
&\quad - \mu \int_\Omega u^\eta (u+1)^{p-1}dx \nonumber\\
 &\leq  -\frac{p-1}{2}\int_\Omega(u+1)^{m+p-3}|\nabla u|^2dx
 +(p-1)\chi^2\int_\Omega(u+1)^{p-m+1}|\nabla v|^2dx \nonumber\\
 &\quad +(p-1)\xi^2\int_\Omega(u+1)^{p-m+1}|\nabla w|^2dx
 +r\int_\Omega (u+1)^{p}dx \nonumber \\
&\quad - \mu \int_\Omega u^\eta (u+1)^{p-1}dx,\quad
\forall t\in(0,T_{\rm max}),\label{Prf3-38}
 \end{align}
 where we have used Young's inequality
in the last step. According to \eqref{Prf3-22} and \eqref{Prf3-23},
we note that there exists a constant $C_1>0$ such that
$\|\nabla v\|^2_{L^4(\Omega)}\leq C_1$,
$\|\nabla w\|^2_{L^4(\Omega)}\leq C_1$ for all $t\in(0, T_{\rm max})$.
Thus, by using H\"{o}lder's inequality and the inequality
$-u^\eta\leq -\frac{(u+1)^\eta}{2^\eta}+1$, we obtain that
\begin{equation}\label{Prf3-39}
\begin{split}
&\frac{d}{dt} \int_\Omega (u+1)^pdx\\
& \leq -\frac{2p(p-1)}{(m+p-1)^2}\int_\Omega |\nabla (u+1)^{\frac{m+p-1}{2}}|^2dx\\
&\quad + p(p-1)(\chi^2+\xi^2)C_1\|(u+1)^{p-m+1}\|_{L^2(\Omega)}\\
 &\quad +pr\int_\Omega (u+1)^{p}dx - \frac{\mu p}{2^\eta}
\int_\Omega (u+1)^{p+\eta-1}dx+\mu p \int_\Omega (u+1)^{p-1}dx
 \end{split}
\end{equation}
for all $t\in(0,T_{\rm max})$. Using that $p>\frac{3m-1}{2}$ implies
\[
\frac{2(m+1)}{m+p-1}<\frac{4(p-m+1)}{m+p-1}.
\]
We note that
\[
\frac{4(p-m+1)}{m+p-1}<6,
\]
because $6(m+p-1)-4(p-m+1)=10(m-1)+2p>0$. We may invoke the
 Gagliardo-Nirenberg inequality to estimate
\begin{align*} %\label{Prf3-40}
 \|(u+1)^{p-m+1}\|_{L^2(\Omega)}
 &= \|(u+1)^{\frac{m+p-1}{2}}\|^{\frac{2(p-m+1)}{m+p-1}}
 _{L^{\frac{4(p-m+1)}{m+p-1}}(\Omega)}\\
 &\leq  C_2\|\nabla(u+1)^{\frac{m+p-1}{2}}\|^{\frac{2(p-m+1)}{m+p-1}
 \theta}_{L^2(\Omega)}
 \|(u+1)^{\frac{m+p-1}{2}}\|^{\frac{2(p-m+1)}{m+p-1}
 (1-\theta)}_{L^{\frac{2(m+1)}{m+p-1}}(\Omega)} \\
 &\quad +C_2 \|(u+1)^{\frac{m+p-1}{2}}\|
 ^{\frac{2(p-m+1)}{m+p-1}}_{L^{\frac{2(m+1)}{m+p-1}}(\Omega)}
 \end{align*}
for all $t\in(0, T_{\rm max})$ with some $C_2>0$ and $\theta \in(0,1)$
determined by
\[
-\frac{3(m+p-1)}{4(p-m+1)}=(1-\frac{3}{2})\theta
-\frac{3(m+p-1)}{2(m+1)}(1-\theta);
\]
that is,
\[
\theta= \frac{\frac{3(m+p-1)}{2}(\frac{1}{m+1}-\frac{1}{2(p-m+1)})}
{\frac{3(m+p-1)}{2(m+1)}-\frac{1}{2}}.
\]
In view of \eqref{Prf3-21}, we obtain that there exists a constant
$C_3>0$ such that
\begin{equation}\label{Prf3-41}
\|(u+1)^{p-m+1}\|_{L^2(\Omega)}\,\,
\leq\,\, C_3\|\nabla(u+1)^{\frac{m+p-1}{2}}\|
^{\frac{2(p-m+1)}{m+p-1}\theta}_{L^2(\Omega)}  +C_3
 \quad \forall t\in(0, T_{\rm max}).
\end{equation}
It is easy to check that
\[
\frac{2(p-m+1)}{m+p-1}\theta=2\frac{3p-\frac{9}{2}m+\frac{3}{2}}{3p+2m-4}<2,
\]
because $(3p+2m-4)-(3p-\frac{9}{2}m+\frac{3}{2})=m+\frac{11}{2}(m-1)>0$.
Thus, by applying Young's inequality, we have
\begin{equation}\label{Prf3-42}
\begin{aligned}
&p(p-1)(\chi^2+\xi^2)C_1\|(u+1)^{p-m+1}\|_{L^2(\Omega)} \\
&\leq \frac{2p(p-1)}{(m+p-1)^2}
\int_\Omega |\nabla (u+1)^{\frac{m+p-1}{2}}|^2dx +C_4
\end{aligned}
\end{equation}
for all $t\in(0, T_{\rm max})$ with some $C_4>0$.
Substituting \eqref{Prf3-42} into \eqref{Prf3-39} yields that
\begin{equation}\label{Prf3-43}
\begin{aligned}
&\frac{d}{dt} \int_\Omega (u+1)^pdx +\int_\Omega (u+1)^pdx \\
& \leq  (pr+1)\int_\Omega (u+1)^{p}dx - \frac{\mu p}{2^\eta}
 \int_\Omega (u+1)^{p+\eta-1}dx\\
&\quad +\mu p \int_\Omega (u+1)^{p-1}dx+C_4
\end{aligned}
\end{equation}
for all $t\in(0, T_{\rm max})$. Using Young's inequality twice again,
there exist some positive constants $C_5$ and $C_6$ such that
\begin{gather}\label{Prf3-44}
(pr+1)\int_\Omega (u+1)^{p}dx \leq \frac{\mu p}{2^{\eta+1}}
\int_\Omega (u+1)^{p+\eta-1}dx+ C_5
\quad \forall t\in(0, T_{\rm max}), \\
\label{Prf3-45}
\mu p \int_\Omega (u+1)^{p-1}dx \leq \frac{\mu p}{2^{\eta+1}}
\int_\Omega (u+1)^{p+\eta-1}dx+ C_6
\quad \forall t\in(0, T_{\rm max}).
\end{gather}
Substituting \eqref{Prf3-44} and \eqref{Prf3-45} into \eqref{Prf3-43}, we have
\[
\frac{d}{dt} \int_\Omega (u+1)^pdx +\int_\Omega (u+1)^pdx  \leq  C_7
\quad \forall t\in(0, T_{\rm max})
\]
with $C_7:=C_4+C_5+C_6$. We thus conclude that there exists $C_8>0$ such that
\[
\int_\Omega (u+1)^pdx\leq C_8\quad \forall t\in(0, T_{\rm max}),
\]
which implies \eqref{Prf3-35}.

From the variation-of-constants representation of $v$, we have
\[
v(\cdot,t)=e^{t(\Delta-\beta)}v_0+\alpha\int^t_0 e^{(t-s)(\Delta-\beta)}u(\cdot,s)ds
\quad \forall t\in(0,T_{\rm max}),
\]
where $(e^{t\Delta})_{t\geq 0}$ is the Neumann heat semigroup in $\Omega$.
Using the $L^p-L^q$ estimate for the Neumann heat semigroup, we can find $C_9>0$
such that
\begin{equation}\label{Prf3-46}
\begin{aligned}
&\|\nabla v(\cdot, t)\|_{L^q(\Omega)} \\
&\leq C_9\|\nabla v_0\|_{L^q(\Omega)}
 +C_2\int^t_0 (t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{p}
 -\frac{1}{q})}e^{-\beta(t-s)}\|u(\cdot,s)\|_{L^p(\Omega)}ds
\end{aligned}
 \end{equation}
 for all $t\in(0, T_{\rm max})$. We note that
$\|u(\cdot, t)\|_{L^P(\Omega)}\leq C_{10}$ with $C_{10}>0$ for all
$t\in(0, T_{\rm max})$ and $p>\frac{3m-1}{2}$.
 In particularly, if we take $p>3$, then we have
 $\frac{1}{2}+\frac{3}{2}(\frac{1}{p}-\frac{1}{q})<1$ for all $q>1$.
Thus we can find a constant $C_{11}>0$ such that
 \begin{equation}\label{Prf3-47}
 \begin{split}
 \|\nabla v(\cdot, t)\|_{L^q(\Omega)}\,\,
 &\leq C_9\|\nabla v_0\|_{L^q(\Omega)}
 +C_2C_{10}\int^t_0 (t-s)^{-\frac{1}{2}
 -\frac{3}{2}(\frac{1}{p}-\frac{1}{q})}e^{-\beta(t-s)}ds\\
 &\leq  C_{11} \quad \forall t\in(0,T_{\rm max}).
 \end{split}
 \end{equation}
 Similarly, we have $\|\nabla w(\cdot, t)\|_{L^q(\Omega)}\leq C_{12}$
for all $t\in(0, T_{\rm max})$ with some constant $C_{12}>0$.
\end{proof}

 In the following subsection, we will deal with the case that the diffusion
is strong enough (i.e., $m>\frac{4}{3}$). We can also establish the $L^p$
estimate for $u$ and $L^{2q}$ estimate for $\nabla v$ and $\nabla w$
for all $p>1$ and $q>1$.

 \subsection*{A priori estimates for the sufficiently strong diffusion case}

\begin{lemma}\label{lem36}
Suppose that $m>\frac{4}{3}$. Let $(u,v,w)$ be a solution to system \eqref{I1}.
Then for all $p>1$ and each $q>1$ there exists a constant $C>0$ such that
\begin{gather}\label{Prf3-48}
\|u(\cdot,t)\|_{L^p(\Omega)}\leq C \quad \forall  t\in(0,T_{\rm max}),\\
\label{Prf3-49}
\|\nabla v(\cdot,t)\|_{L^{2q}(\Omega)}\leq C \quad \forall  t\in(0,T_{\rm max}), \\
\label{Prf3-50}
\|\nabla w(\cdot,t)\|_{L^{2q}(\Omega)}\leq C \quad \forall  t\in(0,T_{\rm max}).
\end{gather}
\end{lemma}

\begin{proof}
 Multiplying the first equation in \eqref{I1} by $(u+1)^{p-1}(p>1)$ and
integrating it over $\Omega$ and using Young's inequality, we obtain that
\begin{equation}\label{Prf3-51}
 \begin{split}
&\frac{1}{p}\frac{d}{dt}\int_\Omega (u+1)^pdx\\
&\leq  -\frac{p-1}{2}\int_\Omega(u+1)^{m+p-3}|\nabla u|^2dx
 +(p-1)\chi^2\int_\Omega(u+1)^{p-m+1}|\nabla v|^2dx\\
&\quad +(p-1)\xi^2\int_\Omega(u+1)^{p-m+1}|\nabla w|^2dx
 +r\int_\Omega (u+1)^{p}dx\\\
&\quad - \mu \int_\Omega u^\eta (u+1)^{p-1}dx
 \end{split}
\end{equation}
for all $t\in(0, T_{\rm max})$. Since $u^\eta \geq \frac{1}{2^\eta}(u+1)^\eta-1$
and $\eta>1$, by using Young's inequality we obtain
\begin{equation}\label{Prf3-52}
 \begin{split}
&r \int_\Omega (u+1)^pdx-\mu \int_\Omega u^\eta (u+1)^{p-1}dx\\
&\leq  r \int_\Omega (u+1)^pdx+ \mu \int_\Omega (u+1)^{p-1}dx
 -\frac{\mu}{2^\eta} \int_\Omega (u+1)^{p+\eta-1}dx \\
&\leq \frac{\mu}{2^{\eta+1}} \int_\Omega (u+1)^{p+\eta-1}dx
 +C_1+\frac{\mu}{2^{\eta+1}} \int_\Omega (u+1)^{p+\eta-1}dx+C_2 \\
&\quad -\frac{\mu}{2^\eta} \int_\Omega (u+1)^{p+\eta-1}dx \\
&= C_1+C_2 \quad \forall t\in(0,T_{\rm max})
 \end{split}
\end{equation}
with some positive constants $C_1$ and $C_2$. Combining \eqref{Prf3-51}
with \eqref{Prf3-52} yields that
\begin{equation}\label{Prf3-53}
 \begin{split}
\frac{1}{p}\frac{d}{dt}\int_\Omega (u+1)^pdx
&\leq  -\frac{2(p-1)}{(m+p-1)^2}\int_\Omega |\nabla(u+1)^{\frac{m+p-1}{2}}|^2dx\\
&\quad + (p-1)\chi^2\int_\Omega (u+1)^{p-m+1}|\nabla v|^2dx\\
&\quad +(p-1)\xi^2\int_\Omega (u+1)^{p-m+1}|\nabla w|^2dx+C_3
 \end{split}
\end{equation}
for all $t\in(0,T_{\rm max})$,
with $C_3:=C_1+C_2$. In Lemma \ref{lem32}, we  obtained the  identity
\[
(|\nabla v|^2)_t=\Delta |\nabla v|^2-2|D^2v|^2
-2\beta|\nabla v|^2+2\alpha\nabla u\cdot \nabla v \quad \forall
t\in(0, T_{\rm max}).
\]
Testing this against $|\nabla v|^{2q-2}$ yields
\begin{equation}\label{Prf3-54}
\begin{split}
 &\frac{1}{q}\frac{d}{dt}\int_\Omega |\nabla v|^{2q}dx
 +\frac{4(q-1)}{q^2}\int_\Omega \big|\nabla |\nabla v|^q\big|^2dx
 +2\int_\Omega |D^2v|^2|\nabla v|^{2q-2}\\
&+2\beta \int_\Omega |\nabla v|^{2q}dx\\
 &\leq \int_{\partial\Omega}|\nabla v|^{2q-2}
 \frac{\partial |\nabla v|^2}{\partial\nu}dx
+ 2\alpha \int_\Omega |\nabla v|^{2q-2}\nabla u\cdot\nabla vdx
 \quad \forall t\in(0,T_{\rm max}),
\end{split}
\end{equation}
where we have used the identity
\[
(q-1)\int_\Omega |\nabla v|^{2q-4}\big|\nabla |\nabla v|^2\big|^2dx
=\frac{4(q-1)}{q^2}\int_\Omega \big|\nabla |\nabla v|^q\big|^2dx.
\]
 For the first integral on the right of \eqref{Prf3-54}, from \cite[(3.10)]{ISY}
we have
\begin{equation}\label{Prf3-55}
 \int_{\partial\Omega}|\nabla v|^{2q-2}\frac{\partial |\nabla v|^2}{\partial\nu}dx
 \leq \frac{q-1}{2q^2}\int_\Omega \big|\nabla |\nabla v|^q\big|^2dx +C_4
 \quad \forall t\in(0,T_{\rm max})
\end{equation}
with some constant $C_4>0$. For the second integral on the right of
\eqref{Prf3-54}, we integrate by parts over $\Omega$ and use Young's
inequality to derive
\begin{equation}\label{Prf3-56}
 \begin{split}
&2\alpha \int_\Omega |\nabla v|^{2q-2}\nabla u\cdot\nabla v\,dx\\
 &=-2\alpha(q-1)\int_\Omega u|\nabla v|^{2q-4}\nabla v\cdot \nabla |\nabla v|^2dx
 -2\alpha \int_\Omega u |\nabla v|^{2q-2}\Delta vdx\\
 &\leq \frac{q-1}{2}\int_\Omega |\nabla v|^{2q-4}\big|\nabla |\nabla v|^2\big|^2dx
 +2\alpha^2(q-1)\int_\Omega u^2|\nabla v|^{2q-2}dx\\
 &\quad +\frac{2}{3}\int_\Omega |\nabla v|^{2q-2}|\Delta v|^2dx
 +\frac{3\alpha^2}{2}\int_\Omega u^2|\nabla v|^{2q-2}dx\\
 \end{split}
\end{equation}
for all $t\in(0,T_{\rm max})$. Upon the pointwise inequality
$|\Delta v|^2\leq 3|D^2v|^2$, we have
\[
\frac{2}{3}\int_\Omega |\nabla v|^{2q-2}|\Delta v|^2dx
\leq 2\int_\Omega |\nabla v|^{2q-2}|D^2v|^2dx
\quad \forall t\in(0,T_{\rm max}).
\]
We thus infer from \eqref{Prf3-54}-\eqref{Prf3-56} that
\begin{equation}\label{Prf3-57}
\begin{aligned}
&\frac{1}{q}\frac{d}{dt}\int_\Omega |\nabla v|^{2q}dx
+\frac{3(q-1)}{2q^2}\int_\Omega \big|\nabla |\nabla v|^q\big|^2dx \\
&\leq (2q-\frac{1}{2})\alpha^2 \int_\Omega u^2|\nabla v|^{2q-2}dx+C_4
\end{aligned}
\end{equation}
 for all $t\in(0,T_{\rm max})$.
By taking a similar procedure for $w$, we derive that
\begin{equation}\label{Prf3-58}
\begin{aligned}
&\frac{1}{q}\frac{d}{dt}\int_\Omega |\nabla w|^{2q}dx
+\frac{3(q-1)}{2q^2}\int_\Omega \big|\nabla |\nabla w|^q\big|^2dx \\
&\leq (2q-\frac{1}{2})\gamma^2 \int_\Omega u^2|\nabla w|^{2q-2}dx+C_5
\end{aligned}
\end{equation}
 for all $t\in(0,T_{\rm max})$,
with some constant $C_5>0$. Thus, collecting \eqref{Prf3-53}, \eqref{Prf3-55}
and \eqref{Prf3-56} yields that
\begin{equation}\label{Prf3-59}
 \begin{split}
 &\frac{d}{dt}\Big\{\frac{1}{p}\int_\Omega (u+1)^pdx
 +\frac{1}{q}\int_\Omega |\nabla v|^{2q}dx
 +\frac{1}{q}\int_\Omega |\nabla w|^{2q}dx\Big\} \\
& +\frac{2(p-1)}{(m+p-1)^2}\int_\Omega |\nabla(u+1)^{\frac{m+p-1}{2}}|^2dx\\
& +\frac{3(q-1)}{2q^2}\int_\Omega \big|\nabla |\nabla v|^q\big|^2dx
 +\frac{3(q-1)}{2q^2}\int_\Omega \big|\nabla |\nabla w|^q\big|^2dx\\
&\leq  (p-1)\chi^2\int_\Omega (u+1)^{p-m+1}|\nabla v|^2dx
 +(p-1)\xi^2\int_\Omega (u+1)^{p-m+1}|\nabla w|^2dx\\
&\quad +(2q-\frac{1}{2})\alpha^2 \int_\Omega (u+1)^2|\nabla v|^{2q-2}dx \\
&\quad +(2q-\frac{1}{2})\gamma^2 \int_\Omega (u+1)^2|\nabla w|^{2q-2}dx+ C_6
 \end{split}
\end{equation}
for all $t\in(0, T_{\rm max})$ with $C_6:=C_4+C_5$.
Note that \eqref{Prf3-59} is similar to \cite[(3.14)]{TW1} and our
condition $m>\frac{4}{3}$ satisfies the condition $1-(m-1)<\frac{2}{N}\,(N=3)$
in \cite{TW1}. Thus, the remaining computation shall follow closely with
minor modification. We omit it for brevity and easily establish
\eqref{Prf3-48}-\eqref{Prf3-50}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1}]
 Using \cite[Lemma A.1]{TW1} and Lemma \ref{lem35} and Lemma \ref{lem36},
 we obtain that there exists a positive constant $C>0$ such that
\[
\|u(\cdot,t)\|_{L^\infty(\Omega)}\leq C \quad \forall t\in(0,T_{\rm max}),
\]
which together with the extensibility criterion \eqref{Pre1} yields
that $T_{\rm max}=+\infty$.
By well-known arguments from parabolic regularity theory for the second
and third equations in \eqref{I1}, we can find some constants $C_2>0$
and $C_3>0$ such that
\begin{gather*}
\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C_2 \quad \forall t>0, \\
\|w(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C_3 \quad \forall t>0.
\end{gather*}
Thus, we prove that $(u,v,w)$ is a global bounded classical solution
to \eqref{I1}.
\end{proof}

 \section{Asymptotic behavior}

 In this section, according to the ideas from \cite[Section 5]{T-W-ZAMP},
we consider the large time behavior of $(u,v,w)$ under the assumptions $r=0$,
$\eta\geq 2$ and $\mu>\max\big\{1+4\chi^2+11\alpha^2+2\chi^2(9+\alpha^2+\gamma^2)$,
$1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2) \big\}$.

 \begin{lemma}\label{lem41}
 Let $r=0$. Suppose that $\eta\geq 2$ and
$\mu>\max\big\{1+4\chi^2+11\alpha^2+2\chi^2(9+\alpha^2+\gamma^2)$,
$1+4\xi^2+11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2) \big\}$.
Then the solution of \eqref{I1} satisfies
 \begin{equation}\label{Ab4-1}
 \int^\infty_0 \int_\Omega u^\eta(x,t)\,dx\,dt< \frac{1}{\mu}\int_\Omega u_0(x)dx
 \quad \forall t>0
 \end{equation}
 and  there exists a constant $C>0$ such that
 \begin{equation}\label{Ab4-2}
 \int_\Omega u(x,t)dx\leq \frac{C}{(t+1)^{\frac{1}{\eta-1}}}  \quad \forall t>0.
 \end{equation}
 \end{lemma}

 \begin{proof}
We integrate the first equation in \eqref{I1} in space to obtain
 \begin{equation}\label{Ab4-3}
 \frac{d}{dt}\int_\Omega u(x,t)dx
=-\mu \int_\Omega u^\eta(x,t) dx \quad \forall t>0.
 \end{equation}
 Integrating it with respect to $t$ yields
 \[
 \int_\Omega u(x,t)dx+\mu \int^t_0\int_\Omega u^\eta(x,t)\,dx\,dt
\leq \int_\Omega u_0(x)dx  \quad \forall t>0,
 \]
 which implies \eqref{Ab4-1}.
 Using H\"{o}lder's inequality, we have
 \[
\int_\Omega udx\leq \big(\int_\Omega u^\eta dx
\big)^{\frac{1}{\eta}}|\Omega|^{1-\frac{1}{\eta}}\]
 and then obtain
$-\int_\Omega u^\eta dx\leq -\frac{1}{|\Omega|^{\eta-1}}
\left(\int_\Omega udx\right)^\eta$.
Substituting it into \eqref{Ab4-3} yields
 \[
 \frac{d}{dt}\int_\Omega u(x,t)dx
 \leq -\frac{\mu}{|\Omega|^{\eta-1}}\Big(\int_\Omega udx\Big)^\eta \quad
\forall t>0.
 \]
 By integrating in time we have
 \[
 \int_\Omega u(x,t)dx\leq
\Big(\frac{1}{\frac{1}{\big(\int_\Omega u_0(x)dx\big)^{\eta-1}}
+\frac{\mu(\eta-1)}{|\Omega|^{\eta-1}}t}\Big)^{\frac{1}{\eta-1}}
 \quad \forall t>0,
 \]
 which implies \eqref{Ab4-2}.
\end{proof}

 \begin{lemma}\label{lem42}
 Let the assumptions in Lemma \ref{lem41} hold. Suppose
that $\beta>\frac{1}{2(\eta-1)}$ and $\delta>\frac{1}{2(\eta-1)}$.
Then there exists $C>0$ such that
 \begin{gather}\label{Ab4-4}
 \int_\Omega v(x,t)dx\leq \frac{C}{(t+1)^{\frac{1}{\eta-1}}}
 \quad \forall t>0, \\
\label{Ab4-5}
 \int_\Omega w(x,t)dx\leq \frac{C}{(t+1)^{\frac{1}{\eta-1}}}
 \quad \text{for\,\,all} \,\,t>0.
 \end{gather}
 \end{lemma}

 \begin{proof}
Integrating the second equation in \eqref{I1} and letting
 $y(t):=\int_\Omega v(x,t)dx$, we see that there exists a constant
 $C_1>0$ such that
\[
 y'(t)= -\beta y(t)+\alpha \int_\Omega u(x,t)dx
 \leq  -\beta y(t)+\frac{\alpha C_1}{(t+1)^{\frac{1}{\eta-1}}}
 \quad \forall t>0.
\]
  We define
 \begin{gather*}
 C_2:=\max\big\{ 2^{\frac{1}{\eta-1}}\int_\Omega v_0(x)dx,\;
\frac{2^{1+\frac{1}{\eta-1}} \alpha C_1}{2\beta-\frac{1}{\eta-1}}\big\}, \\
  \overline{y}(t):=\frac{C_2}{(t+2)^{\frac{1}{\eta-1}}}
 \quad \forall  t>0.
 \end{gather*}
 Then $\overline{y}(0)=\frac{C_2}{2^{\frac{1}{\eta-1}}}
\geq \int_\Omega v_0(x)dx=y(0)$ and
\begin{align*}
&\overline{y}'(t)+\beta \overline{y}(t)
 - \frac{\alpha C_1}{(t+1)^{\frac{1}{\eta-1}}}\\
&= -\frac{C_2}{(\eta-1)(t+2)^{1+\frac{1}{\eta-1}}}
 +\frac{\beta C_2}{(t+2)^{\frac{1}{\eta-1}}}
 -\frac{\alpha C_1}{(t+1)^{\frac{1}{\eta-1}}}\\
&=  \frac{C_2}{(\eta-1)(t+2)^{\frac{1}{\eta-1}}}\big(\frac{1}{2}-\frac{1}{t+2}\big) \\
&\quad +\frac{1}{2(t+2)^{\frac{1}{\eta-1}}}
\Big[(2\beta-\frac{1}{\eta-1})C_2
 -2\big(\frac{t+2}{t+1}\big)^{\frac{1}{\eta-1}}\alpha C_1\Big]\\
&\geq  \frac{C_2}{(\eta-1)(t+2)^{\frac{1}{\eta-1}}}
\big(\frac{1}{2}-\frac{1}{2}\big)
 +\frac{1}{2(t+2)^{\frac{1}{\eta-1}}}
\big[(2\beta-\frac{1}{\eta-1})C_2-2^{1+\frac{1}{\eta-1}}\alpha C_1\big]\\
&\geq  0  \quad \forall t>0,
 \end{align*}
where we have used the assumption $\beta>\frac{1}{2(\eta-1)}$ and the definition
of $C_2$. By comparison, we thus establish \eqref{Ab4-4}.
By taking a similar procedure for $w$, we can also obtain \eqref{Ab4-5}.
\end{proof}

 \begin{lemma}\label{lem43}
 Let $\eta\geq 2$ and $\mu>\max\big\{1+\chi^2+11\alpha^2+2\chi^2
(9+\alpha^2+\gamma^2), 1+\xi^2+ \\
11\gamma^2+2\xi^2(9+\alpha^2+\gamma^2)\big\}$.
Then there exists $\theta \in(0,1)$ and a constant $C>0$ such that
 \begin{equation}\label{AB4-6}
 \|u\|_{C^{\theta,\frac{\theta}{2}}(\overline{\Omega}\times[t,t+1])}\leq C
 \quad \forall t>1.
 \end{equation}
 \end{lemma}

 \begin{proof}
We write the first equation in \eqref{I1} in the form
 \[
 u_t-\nabla\cdot a(x,t,\nabla u)=b(x,t)  \quad x\in\Omega,\; t>0,
 \]
 where $a(x,t,\nabla u):=(u+1)^{m-1}\nabla u-h(x,t)$,
$h(x,t):=\chi u\nabla v-\xi u \nabla w$ and
 $b(x,t):=ru(x,t)-\mu u^\eta(x,t)$ for $x\in\Omega$ and $t>0$.
Here we note that
 \begin{gather*}
 a(x,t,\nabla u)\cdot \nabla u=(u+1)^{m-1}|\nabla u|^2-h\cdot \nabla u
\geq \frac{1}{2}|\nabla u|^2-\frac{1}{2}|h|^2, \\
 |a(x,t,\nabla u)|\leq C_1|\nabla u|+|h|
 \end{gather*}
 in $\Omega\times (0,\infty)$ with $C_1:=(M+1)^{m-1}$,
 where we have used \eqref{I3}. According to Lemma \ref{lem35},
we obtain that $h$ and $b$ belong to $L^\infty((0,\infty); L^q(\Omega))$
for any $q\in(1,\infty)$. By parabolic H\"{o}lder regularity
\cite[Theorem 1.3]{P-V-JDE}, this implies \eqref{AB4-6}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
 Let us assume that the first claim in \eqref{I4} does not hold.
Then we can find a sequence $(t_j)_{j\in \mathbb{N}}\subset (1,\infty)$
and a constant $C_1>0$ such that $t_j\to \infty$ as $j\to \infty$ and
 \begin{equation}\label{Ab4-7}
 \|u(\cdot, t_j)\|_{L^\infty(\Omega)}\geq C_1
 \quad \forall j\in \mathbb{N}.
 \end{equation}
In view of Lemma \ref{lem43} and the Arzel\`{a}-Ascoli theorem,
 we see that $(u(\cdot, t_j))_{j\in \mathbb{N}}$ is relatively compact
in $C^0(\overline{\Omega})$. We can extract a subsequence
 (still denoted by $(u(\cdot, t_j))_{j\in \mathbb{N}}$) such that
\[
u(\cdot, t_j)\to u_\infty \quad \text{in }
L^\infty(\Omega)\quad \text{as} \,\,j\to \infty
\]
with a certain nonnegative $u_\infty\in C^0(\overline{\Omega})$.
 However, from Lemma \ref{lem41} we can obtain that $u(\cdot, t)\to 0$
in $L^1(\Omega)$ as $t\to \infty$. Therefore, we have $u_\infty\equiv 0$,
which contradicts \eqref{Ab4-7}. Thus, we prove
$\|u(\cdot, t)\|_{L^\infty(\Omega)}\to 0 $ as $t\to \infty$.
Upon Lemma \ref{lem35} and Lemma \ref{lem42}, we can take a similar arguments
to prove $\|v(\cdot, t)\|_{L^\infty(\Omega)}\to 0$ and
$\|w(\cdot, t)\|_{L^\infty(\Omega)}\to 0$ as $t\to \infty$.
\end{proof}

\subsection*{Acknowledgements}
The author is very grateful to the referees for their detailed comments
 and valuable suggestions, which greatly improved the manuscript.
This work was partially supported by the Natural Science Project
 of Sichuan Province Department of Education (No.16ZB0075).

\begin{thebibliography}{00}

 \bibitem{ES} E. Espejo, T. Suzuki;
 \emph{Global existence and blow-up for a system describing the
aggregation of microglia}, Appl. Math. Lett. 35 (2014), 29--34.

 \bibitem{E-S-NA} E. Espejo, T. Suzuki;
\emph{Reaction terms avoiding aggregation in slow fluids},
Nonlinear Anal. Real World Appl. 21 (2015), 110--126.

\bibitem{ISY} S. Ishida, K. Seki, T. Yokota;
\emph{Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic
type on non-convex bounded domains}, J. Differential Equations 256 (2014),
2993--3010.

\bibitem{Jin} H. Y. Jin;
\emph{Boundedness of the attraction-repulsion Keller-Segel system},
J. Math. Anal. Appl. 422 (2015), 1463-1478.

\bibitem{JinLiu} H. Y. Jin and Z. Liu;
\emph{Large time behavior of the full attraction-repulsion Keller-Segel
system in whole space}, Appl. Math. Lett. 47 (2015), 13--20.

\bibitem{JW} H. Y. Jin, Z. A. Wang;
\emph{Asymptotic dynamics of the one-dimensional attraction-repulsion
Keller-Segel model}, Math. Methods Appl. Sci. 38 (2015), 444--457.

\bibitem{JinWang} H. Y. Jin, Z. A. Wang;
\emph{Boundedness, blowup and critical mass phenomenon in competing chemotaxis},
J. Differential Equations 260 (2016), 162--196.

\bibitem{K-R-CPDE} A. Kiselev and L. Ryzhik;
\emph{Biomixing by chemotaxis and enhancement of biological reactions},
Commun. Partial Differential Equations 37 (2012), 298--312.

\bibitem{KS} E. F. Keller, L. A. Segel;
\emph{Initiation of slime mold aggregation viewed as an instability},
J. Theor. Biol. 26 (1970), 399--415.

\bibitem{Li-Xie} X. Li;
\emph{Boundedness in a two-dimensional attraction-repulsion system with
nonlinear diffusion}, Math. Methods Appl. Sci. 39 (2016), 289--301.

\bibitem{LCE} M. Luca, A. Chavez-Ross, L. Edelstein-Keshet, A. Mogilner;
\emph{Chemotactic signalling, microglia, and Alzheimers disease senile plague:
Is there a connection? } Bull. Math. Biol. 65 (2003), 673--730.

\bibitem{Li-Lin-Mu} Y. Li, K. Lin, C. Mu;
\emph{Asymptotic behavior for small mass in an attraction-repulsion chemotaxis
system}, Electron. J. Differential Equations 146 (2015), 1--13.

\bibitem{LMW} K. Lin, C. L. Mu, L. C. Wang;
\emph{Large time behavior for an attraction-repulsion chemotaxis system},
J. Math. Anal. Appl. 426 (2015), 105--124.

\bibitem{LSW} P. Liu, J. Shi, Z. A. Wang;
\emph{Pattern formation of the attraction-repulsion Keller-Segel system},
Discrete Cont. Dyn. Systems-B 18 (2013), 2597--2625.

\bibitem{LiuTao} D. Liu, Y. Tao;
\emph{Global boundedness in a fully parabolic attraction-repulsion chemotaxis model},
Math. Methods Appl. Sci. 38 (2015), 2537--2546.

\bibitem{LW} J. Liu, Z. A. Wang;
\emph{Classical solutions and steady states of an attraction-repulsion chemotaxis
in one dimension}, J. Biol. Dynam. 6 (2012), 31--41.

\bibitem{Li-Xiang2} X. Li, Z. Xiang;
\emph{On an attraction-repulsion chemotaxis system with a logistic source},
IMA J. Appl. Math. 81 (2016), 165--198.

\bibitem{PH} K. J. Painter, T. Hillen;
\emph{Volume-filling and quorum-sensing in models for chemosensitive movement},
Can. Appl. Math. Quart. 10 (2002), 501--543.

\bibitem{P-V-JDE} M. M. Porzio, V. Vespri;
\emph{H\"{o}lder estimates for local solutions of some doubly nonlinear
degenerate parabolic equations}, J. Differential Equations 103 (1993), 146--178.

\bibitem{SSW-SIAM} C. Stinner, C. Surulescu, M. Winkler;
\emph{Global weak solutions in a PDE-ODE system modeling multiscale cancer
cell invasion}, SIAM J. Math. Anal. (46) 2014, 1969--2007.

\bibitem{SW} R. K. Shi, W. K. Wang;
\emph{Well-posedness for a model derived from an attraction-repulsion
chemotaxis system}, J. Math. Anal. Appl. 423 (2015), 497--520.

\bibitem{TW} Y. Tao, Z. A. Wang;
\emph{Competing effects of attraction vs. repulsion in chemotaxis},
Math. Models Methods Appl. Sci. 23 (2013), 1--36.

\bibitem{TW1} Y. Tao, M. Winkler;
\emph{Boundedness in a quasilinear parabolic-parabolic Keller-Segel system
with subcritical sensitivity}, J. Differential Equations 252 (2012), 692--715.

\bibitem{T-W-ZAMP} Y. Tao, M. Winkler;
\emph{Boundedness and decay enforced by quadratic degradation in a
three-dimensional chemotaxis-fluid system}, Z. Angew. Math. Phys. 66 (2015),
2555--2573.

\bibitem{T-W-SIAM} Y. Tao, M. Winkler;
\emph{A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and
logistic source}, SIAM J. Math. Anal. 43 (2011), 685--704.

\bibitem{Wang-BVP} Y. Wang;
\emph{Global existence and boundedness in a quasilinear attraction-repulsion
chemotaxis system of parabolic-elliptic type}, Bound. Value Probl. 2016 (2016),
1--22.

\bibitem{Wang-DCDSB} Y. Wang, Z. Xiang;
\emph{Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion
chemotaxis system}, Discrete Cont. Dyn. Systems-B 21 (2016), 1953--1973.

\bibitem{Wang-JMAA} Y. Wang;
\emph{A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic
type with logistic source}, J. Math. Anal. Appl. 441 (2016), 259--292.

\end{thebibliography}

\end{document}