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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 24, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/24\hfil Two-species competition models]
{Two-species competition models with fitness-dependent dispersal
on non-convex bounded domains} 

\author[X. Li \hfil EJDE-2015/24\hfilneg]
{Xie Li}

\address{Xie Li \newline
School of Mathematical Sciences, University of Electrical
Science \& Technology of China,
Chengdu 611731, China.\newline
College of Mathematic and Information, China West Normal University,
Nanchong 637002, China}
\email{xieli-520@163.com}

\thanks{Submitted September 18, 2014. Published January 27, 2015.}
\subjclass[2000]{35A01, 35B40, 35K57, 92D25}
\keywords{Two-species competition models; global solution;
bounded solution}

\begin{abstract}
 In this article, we show the existence of global bounded  solutions
 to a two-species competition models with fitness-dependent dispersal
 posed in a bounded domain $\Omega\subset\mathbb{R}^N$ with smooth boundary.
 In particular, we remove the convexity assumption on $\Omega$ used
 by Lou-Tao-Winkler \cite{LTW}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


 \section{Introduction}

 In this article,  we show the existence and boundedness of global solutions 
to the  two-species competition model with fitness-dependent dispersal
 \begin{equation}\label{AH}
\begin{gathered}
u_t=\nabla\cdot[\mu\nabla u-\alpha u\nabla(m-u-w)]+r_1u(m-u-w),\quad
  x\in\Omega,\;t>0,\\
w_t=\nu\Delta w+r_2w(m-u-w), \quad x\in\Omega,\;t>0,\\
[\mu\nabla u-\alpha u\nabla(m-u-w)]\cdot n
=\nu\nabla w\cdot n=0,\quad  x\in\partial\Omega,\;t>0,\\
u(x,0)=u_{0}(x),\quad  w(x,0)= w_{0}(x),\quad x\in\Omega,
\end{gathered}
\end{equation}
where  $\mu, \nu,\alpha>0, r_1\ge 0, r_2\ge 0$,  
 $\Omega\subset\mathbb{R}^N$ $(N\ge1)$ is a bounded domain with smooth boundary 
$\partial\Omega$ and $n$ denotes the outer unite normal of $\partial\Omega$. 
The functions $u(x,t)$ and $w(x,t)$ describe the densities of two competing 
species at time $t$, at location $x\in\Omega$,  and $m(x)$ denotes the 
distribution of resources. Equation $\eqref{AH}_1$ indicates that the dispersal 
of organism with density $u$ is dependent on a combination of random motion 
with random dispersal rate $\mu$ and advection upward along its fitness gradient,
 while equation $\eqref{AH}_2$ indicates that the dispersal of organism with 
density $w$ is purely random. Moreover, the growth of both species in 
\eqref{AH} is logistic, with logistic growth rate $r_1$ and $r_2$,  
respectively.

    In recent years, equations \eqref{AH} and their variations have been 
studied by many researchers (see \cite{A, AR,CCL,CCLX,C,CW,DHMP,H,HMP,KLS, TW2} 
and references therein).  To motivate our study, we recall several related ones. 
Cosner \cite{C} first considered the following fitness-dependent dispersal 
model for a single species
   \begin{equation}\label{AH1}
\begin{gathered}
u_t=\nabla\cdot[\mu\nabla u-\alpha u\nabla(m-u)]+ru(m-u),\quad  
  x\in\Omega,\; t>0,\\
[\mu\nabla u-\alpha u\nabla(m-u)]\cdot n=0,\quad  x\in\Omega,\;t>0.
\end{gathered}
\end{equation}
Then  Cantrell-Cosner-Lou \cite{CCL} further investigated the global existence 
of classical solution and  the behavior of equilibria to equation \eqref{AH1}. 
Recently,  Cantrell et al  \cite{CCLX}  extended the work in \cite{C} to the 
 two-species competition model
\begin{equation}\label{AH2}
\begin{gathered}
u_t=\nabla\cdot[\mu\nabla u-\alpha u\nabla(m-u-w)]+ru(m-u-w),\quad 
  x\in\Omega,\; t>0,\\
w_t=\nu\Delta w+rw(m-u-w)\quad  x\in\Omega,\; t>0,\\
[\mu\nabla u-\alpha u\nabla(m-u-w)]\cdot n=\nu\nabla w\cdot n=0,\quad  
x\in\partial\Omega,\; t>0,\\
u(x,0)=u_{0}(x),\quad  w(x,0)= w_{0}(x),\quad x\in\Omega,
\end{gathered}
\end{equation}
and showed that the solutions to equations \eqref{AH2} exist globally 
for $N = 1, 2$, and for $N \ge 3$ with  $\nu>\mu$.  They  also investigated 
the nontrivial nonnegative  steady states. More recently,  Lou et al \cite{LTW} 
proved that the corresponding results hold for $N\ge3$ and 
$\nu\le\mu$  under the extra assumption that  the domain $\Omega$ is convex. 
The global existence and  large time behaviour of the nonnegative weak solution 
to the limit case (i.e., $\mu=\nu=0$) were also investigated by \cite{LTW}.

The main purpose of this article is to show the global-in-time existence and 
uniform-in-time boundedness of solutions to  \eqref{AH} on a 
{\it non-convex} bounded domain. For this purpose, we recall two basic
 assumptions used in  \cite{LTW}. The first one is related to the parameters 
and the distribution of resources:
\begin{equation}\label{IC1}
m(x)\in C^{2+\gamma}(\overline{\Omega})\quad\text{for some 
$\gamma\in(0,1)$, and $m(x_0)>0$ for some } x_0\in\Omega.
\end{equation}
The second one relates the initial data:
\begin{equation}\label{IC2}
\begin{gathered}
\text{$(u_{0}, w_0)\in C^{\gamma}(\overline{\Omega})\times W^{1,\infty}(\Omega)$
for  some $\gamma\in(0,1)$,} \\
\text{ and $u_0(x)\ge 0$, $w_{0}(x)\ge 0$ in $\overline{\Omega}$}.
\end{gathered}
\end{equation}
To obtain the uniqueness, we also need the following  conditions:
\begin{equation}\label{IC3}
\begin{gathered}
\text{$(u_0,w_0)\in W^{s,p}(\Omega)\times W^{s,p}(\Omega)$ for  some 
$p>N$ and $s>1$,}\\
\text{and $u_0(x)> 0$, $w_{0}(x)\ge 0$ in $\overline{\Omega}$}.
\end{gathered}
\end{equation}
We now  state the main result of this paper as follows.

\begin{theorem}\label{thm1}
 Let $\Omega\subset\mathbb{R}^N(N\ge 1)$  be a bounded  domain with 
smooth boundary $\partial\Omega$. Then under the assumptions of 
\eqref{IC1} and \eqref{IC2},  equations \eqref{AH} have at least one 
couple of nonnegative classical solutions $(u,w)$ belonging to 
$C^0(\overline{\Omega}\times[0,\infty))\cap C^{2,1}
(\overline{\Omega}\times(0,\infty))$,  which are bounded in 
$\Omega\times (0,\infty)$.  If in addition  $(u_0,w_0)$ also satisfy \eqref{IC3}, then the solution is unique within the indicated class.
\end{theorem}

The rest of this article is organized as follows. 
We first present the local existence and uniqueness of classical solutions, 
and some preliminaries in Section 2.  
Then we establish the global existence of bounded solutions and 
complete the proof of Theorem \ref{thm1} in Section 3.

\section{Preliminaries}

In this section, we first present the  existence and uniqueness of 
classical local solutions to \eqref{AH} and then present 
some basic preliminaries.

\begin{lemma}[Local existence and uniqueness]\label{LE}
Under assumptions \eqref{IC1} and \eqref{IC3}, there exists a maximal existence 
time $T^*$ and a unique functions pair 
$(u,w)\in C^0\big(\overline{\Omega}\times [0,T^*)\big)
\cap  C^{2,1}\big(\overline{\Omega}\times (0,T^*)\big)$ 
such that $(u,w)$ are classical solutions of equations \eqref{AH}.
 Moreover, if $T^*<\infty$, then
\begin{equation}\label{BH}
\lim_{t\to T^*}\|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty.
\end{equation}
\end{lemma}

The proof of the above lemma  is standard and we refer to 
\cite{CCLX,LTW} for details.
The following boundary derivative estimate plays an important 
role when we remove the  convexity assumption on the domain $\Omega$ 
used by \cite{WLM}.

\begin{lemma}[{\cite[Lemma 4.2]{MS}}] \label{lem1}
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with smooth boundary  
$\partial\Omega$. If $f\in C^2(\overline{\Omega})$ satisfies 
$\frac{\partial f}{\partial n}=0$, then
\begin{equation}\label{PH4}
\frac{\partial |\nabla f|^2}{\partial n}\le C_\Omega|\nabla f|^2,
\end{equation}
where $C_\Omega$ is a positive upper bound for the curvatures of $\partial\Omega$.
\end{lemma}

The following embedding theorem  comes from 
\cite[Proposition 4.22 (ii) and Theorem 4.24 (i)]{HT}.

\begin{lemma}\label{lem2}
Let $\Omega$ be a bounded domain  with smooth boundary and let $r\in(0,\infty)$. 
Then
\begin{equation*}
W^{r,2}(\partial\Omega)\hookrightarrow L^2(\partial\Omega)
\end{equation*}
is a compact embedding. Moreover, there exists a linear and bounded map 
from $W^{r+\frac{1}{2},2}(\Omega)$ onto $W^{r,2}(\partial\Omega)$.
\end{lemma}

The proof of global existence will be based on some
a priori estimates. To derive these estimates, we will use the following 
two Gagliardo-Nirenberg inequalities \cite{F,N,TW0}.

\begin{lemma}\label{lem3}
Assume that $u\in W^{1,2}(\Omega)\cap L^r(\Omega)$ and $r\in (0,k)$. 
Then there exists a positive constant $C_{GN}$ such that
\[
\|u\|_{L^k(\Omega)}\le C_{GN}\Big(\|\nabla n\|
^\theta_{L^2(\Omega)}\|u\|^{1-\theta}_{L^r(\Omega)}+\|u\|_{L^r(\Omega)}\Big)
\]
holds, where $\theta\in(0,1)$ satisfies
\[
\frac{1}{k}=\theta\Big(\frac{1}2-\frac1N\Big)+\frac{1}{r}\Big(1-\theta\Big).
\]
\end{lemma}


\begin{lemma}\label{lem4}
Let $N\in\mathbb{N}$, $s\ge 1$ and $l\ge 1$. Assume that $p>0$ and $\theta\in (0,1)$ 
satisfy
\begin{equation}\label{PH5}
\frac{1}{2}-\frac{p}{N}=(1-\theta)\frac{l}{s}
+\theta\Big(\frac{1}{2}-\frac{1}{N}\Big) \quad \text{and} \quad p\le \theta.
\end{equation}
Then there exists  a  positive constant $C_0$  such that
\begin{equation}\label{PH6}
\|f\|_{ W^{p,2}(\Omega)}\le C_0\|\nabla f\|^{\theta}_{L^2(\Omega)}
\|f\|^{1-\theta}_{L^{\frac{s}{l}}(\Omega)}+C_0\|f\|_{L^{\frac{s}{l}}(\Omega)}
\end{equation}
holds  for all $f\in W^{1,2}(\Omega)\cap L^{\frac{s}{l}}(\Omega)$.
\end{lemma}

\section{Proof of Theorem \ref{thm1}}

In this section, we  establish the  existence of classical global solutions 
to  \eqref{AH}. For this purpose, the key is to derive the uniform estimate 
of $L^k$ norm of the solution. 
Inspired by an idea in \cite{TW,TW1} (see also \cite{LTW}), we  establish 
a combined estimate on 
$\int_\Omega u^k(x,t) dx+\int_\Omega |\nabla w(x,t)|^{2l} dx$ 
for appropriately large $k$ and $l$ to obtain the expecting results. 
 We first recall some basic properties of solutions.

\begin{lemma}[{\cite[Lemma 2.2]{LTW}}] \label{L1E}
Assume that $u_0(x)\in C^2(\overline{\Omega})$ is positive and 
$w_0(x)\in C^2(\overline{\Omega})$ is nonnegative. 
Then the classical solution $(u,w)$ to equations \eqref{AH} satisfies 
the following inequalities:
\begin{gather}\label{L1E1}
\|u(\cdot,t)\|_{L^1(\Omega)}
\le \max\{\|u_0\|_{L^1(\Omega)},|\Omega|\|m\|_{L^1(\Omega)}\}, \\
\label{L1E2}
\|w(\cdot,t)\|_{L^1(\Omega)}
\le  \max\{\|w_0\|_{L^1(\Omega)},|\Omega|\|m\|_{L^1(\Omega)}\}, \\
\label{LIE2}
\|w(\cdot,t)\|_{L^\infty(\Omega)}
\le \max\{\|u_0\|_{L^1(\Omega)},|\Omega|\|m\|_{L^\infty(\Omega)}\}
\end{gather}
for  all $t\in (0,T^*)$.  Moreover, for any $s\in[1,\frac{N}{N-1})$, 
there exists a positive constant $C(s)$ such that
\begin{equation}\label{W1sE2}
\|w(\cdot,t)\|_{W^{1,s}(\Omega)}\le C(s)
\Big(1+\|u_0\|_{L^1(\Omega)}+\|w_0\|_{W^{1,\infty}(\Omega)}\Big)\quad 
 \text{for  all } t\in (0,T^*).
\end{equation}
\end{lemma}


The following Lemma asserts the $L^k$-boundedness of solutions,
 which is the core of the argument concerning the global existence 
and boundedness. Our proof followed from \cite[Lemma 2.5]{LTW},
 but we will use the boundary derivative estimates and the Sobolev 
trace embedding to remove the convexity assumption on the domain
 $\Omega$ used by \cite{LTW}.

\begin{lemma}\label{LkE}
Let $\Omega\subset \mathbb{R}^N$ $(N\ge 1)$ be a bounded domain with 
smooth boundary $\partial\Omega$. Assume that $\mu$, $\nu$, $\alpha$, $r_1$, 
$r_2$ and $m(x)$  satisfy \eqref{IC1}.  Then for all $k> 2$ and $l> 2$, 
there exist two positive constants $C_k$ and $C_{2l}$ depending only on 
$k$, $l$, $\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{W^{1,\infty}(\Omega)}$ 
such that the solution $(u,w)$ to equations  \eqref{AH} emanating from 
some initial data $(u_0,w_0)\in C^2(\overline{\Omega})\times C^2(\overline{\Omega})$ 
satisfies 
\begin{gather}\label{LkE1}
\|u(\cdot,t)\|_{L^k(\Omega)}\le C_k \quad \text{for all } t\in(0,T^*),\\
\label{LlE2}
\|\nabla w(\cdot,t)\|_{L^{2l}(\Omega)}\le C_{2l} \quad \text{for all } t\in(0,T^*).
\end{gather}
\end{lemma}

\begin{proof}
  The $L^1$-boundedness of $u(\cdot,t)$ has been obtained in the proof 
of Lemma \ref{L1E1}. Thus by the interpolation, we may pay our attention 
to the case that $k>2$.
 Multiplying \eqref{AH}$_1$ by $ku^{k-1}$ and integrating the resulted 
equation over $\Omega$, we obtain
\begin{equation}\label{G1}
\begin{split}
&\frac{d}{dt}\int_\Omega u^k dx\\
&=-\mu k(k-1)\int_\Omega u^{k-2}|\nabla u|^2 dx
 -\alpha k(k-1)\int_\Omega u^{k-1}|\nabla u|^2 dx\\
&\quad+\alpha k(k-1)\int_\Omega u^{k-1}\nabla u\cdot\nabla m dx
 -\alpha k(k-1)\int_\Omega u^{k-1}\nabla u\cdot\nabla w dx \\
&\quad +kr_1\int_\Omega u^{k}(m-u-w)dx\\
&\le -\alpha k(k-1)\int_\Omega u^{k-1}|\nabla u|^2 dx
 +\alpha k(k-1)\int_\Omega u^{k-1}\nabla u\cdot\nabla m dx\\
 &\quad-\alpha k(k-1)\int_\Omega u^{k-1}\nabla u\cdot\nabla w dx
 +kr_1\|m\|_{L^\infty(\Omega)}\int_\Omega u^{k}dx \quad\forall  
 t\in (0,T^*).
\end{split}
\end{equation}
Then following the same procedure as Step 1 of  \cite[Lemma 2.5]{LTW}, we obtain
\begin{equation}\label{G2}
\begin{aligned}
&\frac{d}{dt}\int_\Omega u^k dx+\int_\Omega u^k dx
 +\frac{\alpha k(k-1)}{(k+1)^2}\int_\Omega \big|\nabla u^{\frac{k+1}2}\big|^2dx\\
&\le \alpha k(k-1)\int_\Omega u^{k-1}|\nabla w|^2 dx+C_1(k,\|u_0\|_{L^\infty(\Omega)},
\|w_0\|_{W^{1,\infty}(\Omega)})
\end{aligned}
\end{equation}
for all $ t\in (0,T^*)$. We  divide the proof into two cases.
\smallskip

\noindent\textbf{Case (i):  $N=1$.}  
First of all,  for any $l\in(2,\infty)$, there exists a positive constant 
$C_{2l}\big(l,\|u_0\|_{L^\infty(\Omega)}, \|w_0\|_{W^{1,\infty}(\Omega)}\big)$ 
such that \eqref{LlE2} holds, i.e.
$\int_\Omega|\nabla w|^{2l}\le C_{2l}$
by  \eqref{W1sE2}.  Next, to estimate $\|u(\cdot,t)\|_{L^k(\Omega)}$, 
we  use the Gagliardo-Nirenberg interpolation inequality (Lemma \ref{lem3}) 
and \eqref{L1E1} to obtain
\begin{align*}
\int_\Omega u^kdx
&=\|u^{\frac{k+1}2}\|^{\frac{2k}{k+1}}_{L^{\frac{2k}{k+1}}{(\Omega)}}\\
&\le C(k)\Big(\|\nabla u^{\frac{k+1}2}\|^\theta_{L^2{(\Omega)}}
 \|u^{\frac{k+1}2}\|^{1-\theta}_{L^{\frac{2}{k+1}}{(\Omega)}}
 +\|u^{\frac{k+1}2}\|_{L^{\frac{2}{k+1}}{(\Omega)}}\Big)^{\frac{2k}{k+1}}\\
&\le C_3\Big(\|\nabla u^{\frac{k+1}2}\|^{\frac{2k\theta}{k+1}}_{L^2{(\Omega)}}
 +1\Big),
\end{align*}
where $\theta=\frac{\frac{k+1}2-\frac{k+1}{2k}}{1-\frac{1}2+\frac{k+1}2}\in (0,1)$, 
and $C_3$ is a positive constant depending only on $k$ and  $\|u\|_{L^1(\Omega)}$.
 A simple computation shows that
$\frac{2k\theta}{k+1}<2$.
It then follows from Young's inequality  and \eqref{LlE2} that
\begin{equation}\label{N11}
\begin{split}
&\int_\Omega u^{k-1}|\nabla w|^2 dx\\
&\le\int_\Omega u^k dx+\int_\Omega |\nabla w|^{2k} dx
\le\int_\Omega u^k dx+C_{2k}\\
&\le C_3\|\nabla u^{\frac{k+1}2}\|^{\frac{2k\theta}{k+1}}_{L^2{(\Omega)}}
 +C_4\left(l,\|u_0\|_{L^\infty(\Omega)}, \|w_0\|_{W^{1,\infty}(\Omega)}\right)\\
&\le\frac1{2(k+1)^2}\int_\Omega \big|\nabla u^{\frac{k+1}2}\big|^2dx
+C_5\left(l,\|u_0\|_{L^\infty(\Omega)}, \|w_0\|_{W^{1,\infty}(\Omega)}\right)
\end{split}
\end{equation}
for all $t\in(0,T^*)$. Combining \eqref{N11} with \eqref{G2}, and using ODE
 comparison argument, we obtain the desired estimate \eqref{LkE1}.
\smallskip

\noindent\textbf{Case (ii):  $N\ge2$.}
 In this case, the estimate \eqref{LlE2} can not be derived from \eqref{W1sE2} 
directly. To overcome this difficulty, we will establish a combined 
estimate on $\int_\Omega u^k(x,t) dx+\int_\Omega |\nabla w(x,t)|^{2l} dx$.
 For this purpose, we differentiate equation \eqref{AH}$_2$ to obtain
$$
\big(|\nabla w|^2\big)_t=2\nu\nabla w\cdot\nabla\Delta w
+2r_2\nabla w\cdot\nabla [w(m-u-w)],
$$
which together with the point-wise identity 
$2\nabla w\cdot\nabla\Delta w=\Delta|\nabla w|^2-2|D^2 w|^2$ yields 
\begin{equation}\label{N21}
\big(|\nabla w|^2\big)_t=\nu\Delta|\nabla w|^2-2\nu|D^2 w|^2
+2r_2\nabla w\cdot\nabla [w(m-u-w)].
\end{equation}
Multiplying both sides of \eqref{N21} by $l|\nabla w|^{2(l-1)}$ and 
integrating over $\Omega$, we have
\begin{equation}\label{N22}
\begin{split}
&\frac{d}{dt}\int_\Omega |\nabla w|^{2l}dx\\
&=\nu l\int_\Omega |\nabla w|^{2(l-1)}\Delta|\nabla w|^2dx
 -2\nu l\int_\Omega |\nabla w|^{2(l-1)}|D^2 w|^2dx\\
&\quad +2r_2 l\int_\Omega |\nabla w|^{2(l-1)}\nabla w\cdot\nabla [w(m-u-w)]dx\\
&=-\nu l(l-1)\int_\Omega |\nabla w|^{2(l-2)}|\nabla|\nabla w|^2|^2dx
 +l\nu\int_{\partial\Omega}|\nabla w|^{2(l-1)}
 \frac{\partial|\nabla w|^2}{\partial n}dx\\
&\quad -2\nu l\int_\Omega |\nabla w|^{2(l-1)}|D^2 w|^2dx \\
&\quad +2r_2 l\int_\Omega |\nabla w|^{2(l-1)}\nabla w\cdot\nabla [w(m-u-w)]dx\quad 
\text{for  all } t\in (0,T^*).
\end{split}
\end{equation}

The estimate for the second term on the right-hand side of \eqref{N22} 
is very  subtle.  We first use Lemma \ref{lem1} to obtain
\begin{equation}\label{N23}
\begin{split}
\int_{\partial\Omega}|\nabla w|^{2(l-1)}\frac{\partial
|\nabla w|^2}{\partial n}dx&\le C_\Omega\int_{\partial\Omega}|\nabla w|^{2l}dx
=C_\Omega\big\||\nabla w|^l\big\|^2_{L^2(\partial\Omega)}
\end{split}
\end{equation}
\quad
for all $t\in (0,T^*)$.
Then let us fix a constant $r\in\big(0,\frac12\big)$. Since the embedding 
$W^{r+\frac{1}{2},2}(\Omega)(\hookrightarrow W^{r,2}(\partial\Omega))
\hookrightarrow  L^2(\partial\Omega)$ is compact by Lemma \ref{lem2}, we have
\begin{equation}\label{N24}
\||\nabla w|^l\|_{L^2(\partial\Omega)}
\le C \||\nabla w|^l\|_{W^{r+\frac{1}{2},2}(\Omega)}.
\end{equation}
To estimate the right-hand side of \eqref{N24}, we  take two constants 
$s\in \big[1,\frac{N}{N-1}\big)$ and  $\theta\in\big(0,1\big)$ such that
\[
\frac{1}{2}-\frac{r+\frac1{2}}{N}=(1-\theta)\frac{l}{s}
+\theta\Big(\frac{1}{2}-\frac{1}{N}\Big).
\]
Noticing that $l>1$ implies that $r+\frac1{2}\le \theta<1$, 
we can apply the fractional Gagliardo-Nirenberg inequality 
(Lemma \ref{lem4}) to the right hand side of \eqref{N24} to obtain
\begin{equation}\label{N25}
\begin{split}
\||\nabla w|^l\|_{ W^{r+\frac1{2},2}(\Omega)}
&\le C_0\|\nabla |\nabla w|^l\|^{\theta}_{L^2(\Omega)}
 \||\nabla w|^l\|^{1-\theta}_{L^{\frac{s}{l}}(\Omega)}
 +\tilde{C}_0\||\nabla w|^l\|_{L^{\frac{s}{l}}(\Omega)}\\
&= C_0\|\nabla |\nabla w|^l\|^{\theta}_{L^2(\Omega)}
 \|\nabla w\|^{(1-\theta)l}_{L^{s}(\Omega)}
 +\tilde{C}_0\|\nabla w\|^{l}_{L^{s}(\Omega)}\\
&\le C\Big(\|\nabla |\nabla w|^l\|^{\theta}_{L^2(\Omega)}+1\Big)\quad 
\text{for all } t\in (0,T^*).
\end{split}
\end{equation}
Here we used the boundedness of $\|w(\cdot,t)\|_{W^{1,s}(\Omega)}$ 
in the last inequality (see Lemma \ref{L1E}). Substituting \eqref{N24} 
and \eqref{N25} into \eqref{N23}, and applying Young's inequality with 
$\epsilon$, we have
\begin{equation}\label{N26}
\begin{split}
\int_{\partial\Omega}|\nabla w|^{2l-2}
\frac{\partial|\nabla w|^2}{\partial n}dx
&\le C\Big(\|\nabla |\nabla w|^l\|^{2\theta}_{L^2(\Omega)}+1\Big)
= C\Big(\int_\Omega|\nabla |\nabla w|^l|^2dx\Big)^{\theta}+C\\
&\le \epsilon\int_\Omega|\nabla |\nabla w|^l|^2dx+C(\epsilon)
\end{split}
\end{equation}
for all $t\in (0,T^*)$,  where $\epsilon$ is a positive constant to be 
specified later. For the last term on the right hand of \eqref{N22}, 
we follow the same procedure as \cite[(2.26)--(2.29)]{LTW} and obtain
\begin{equation}\label{N27}
\begin{split}
&2r_2 l\int_\Omega |\nabla w|^{2(l-1)}\nabla w\cdot\nabla [w(m-u-w)]dx\\
&\le2\nu l\int_\Omega |\nabla w|^{2(l-1)}|D^2w|^2dx
 +\frac{(l-1)\nu l}2\int_\Omega |\nabla w|^{2(l-2)}|\nabla|\nabla w|^2|^2dx\\
&\quad +C_4\int_\Omega |\nabla w|^{2(l-1)}dx+C_5\int_\Omega u^2|\nabla w|^{2(l-1)}dx,
\end{split}
\end{equation}
where $C_4$ and $C_5$ are positive constants depending on $l$, 
$\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{W^{1,\infty}(\Omega)}$.
Since
$$
|\nabla w|^{2(l-2)}|\nabla|\nabla w|^2|^2=\frac4{l^2}|\nabla|\nabla w|^l|^2,
$$
we combine \eqref{N26} and \eqref{N27} with \eqref{N22}, and utilize 
Young's inequality to obtain
\begin{equation}\label{N28}
\begin{split}
&\frac{d}{dt}\int_\Omega |\nabla w|^{2l}dx
 +\int_\Omega |\nabla w|^{2l}dx+\Big(\frac{\nu 2(l-1)}l-l\nu\epsilon\Big)
 \int_\Omega|\nabla |\nabla w|^l|^2dx\\
&\le\int_\Omega |\nabla w|^{2l}dx+ C_4\int_\Omega |\nabla w|^{2(l-1)}dx
 +C_5\int_\Omega u^2|\nabla w|^{2(l-1)}dx+C(\epsilon)\\
&\le2\int_\Omega |\nabla w|^{2l}dx
 +C_5\int_\Omega u^2|\nabla w|^{2(l-1)}dx+C(\epsilon,|\Omega|,l)\quad 
\text{for all } t\in (0,T^*).
\end{split}
\end{equation}
For any $s\in\big[1,\min\{\frac{N}{N-1},2l\}\big)$, we take
\[
\tilde{\theta}=\big(\frac{1}{2}-\frac{l}{s}\big)
\big(\frac{1}{2}-\frac{1}{N}-\frac{l}{s}\big)^{-1}.
\]
Then the Gagliardo-Nirenberg inequality gives 
\begin{align*}
\int_\Omega |\nabla w|^{2l}dx
&=\||\nabla w|^l\|^2_{ L^2(\Omega)}\\
&\le C(l)\|\nabla |\nabla w|^l\|^{2\tilde{\theta}}_{L^2(\Omega)}
 \||\nabla w|^l\|^{2(1-\tilde{\theta})}_{L^{\frac{s}{l}}(\Omega)}
 +C(l)\||\nabla w|^l\|^2_{L^{\frac{s}{l}}(\Omega)}\\
&\le C_6\Big(\|\nabla |\nabla w|^l\|^{2\tilde{\theta}}_{L^2(\Omega)}+1\Big)
\end{align*}
for all $t\in (0,T^*)$, where  $C_6$ is a  positive constant depending on
 $l$, $\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{W^{1,\infty}(\Omega)}$. 
Here we used the boundedness of $\|w(\cdot,t)\|_{W^{1,s}(\Omega)}$ in the 
last inequality.
Since $s<2l$, a simple computation shows that $\tilde{\theta}\in(0,1)$, 
i.e, $2\tilde{\theta}<2$. Thus by utilizing Young's inequality, we have
\begin{equation*}
2\int_\Omega |\nabla w|^{2l}dx
\le l\nu\epsilon\int_\Omega|\nabla |\nabla w|^l|^2dx+C(\epsilon).
\end{equation*}
Upon substituting into \eqref{N28}, and taking $\epsilon=\frac{l-1}{l^2}$, 
we obtain
\begin{equation}\label{N29}
\begin{aligned}
&\frac{d}{dt}\int_\Omega |\nabla w|^{2l}dx
 +\int_\Omega |\nabla w|^{2l}dx+\frac{\nu (l-1)}l
 \int_\Omega|\nabla |\nabla w|^l|^2dx \\
&\le C_5\int_\Omega u^2|\nabla w|^{2(l-1)}dx +C_7
\end{aligned}
\end{equation}
 for  all $t\in (0,T^*)$,
where $C_7$ is a positive constant depending on $l$, 
$|\Omega|$, $\|u_0\|_{L^\infty(\Omega)}$ and  $\|w_0\|_{W^{1,\infty}(\Omega)}$.
Now adding \eqref{N29} to \eqref{G2}, we have
\begin{equation}\label{G3}
\begin{split}
&\frac{d}{dt}\Big(\int_\Omega u^k dx+\int_\Omega |\nabla w|^{2l}dx\Big)
+\int_\Omega u^k dx+\int_\Omega |\nabla w|^{2l}dx\\
&+\frac{\alpha k(k-1)}{(k+1)^2}\int_\Omega \big|\nabla u^{\frac{k+1}2}\big|^2dx
+\frac{\nu (l-1)}l\int_\Omega|\nabla |\nabla w|^l|^2dx\\
&\le\alpha k(k-1)\int_\Omega u^{k-1}|\nabla w|^2 dx
 +C_5\int_\Omega u^2|\nabla w|^{2(l-1)}dx+C_8
\end{split}
\end{equation}
for all $t\in (0,T^*)$,
where $C_8$ is a positive constant  depending only  on $k$, $l$, $|\Omega|$, 
$\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{W^{1,\infty}(\Omega)}$. 
Following the same procedure as \cite[(2.35)--(2.43)]{LTW},
 we can find a positive constant $C_9$ depending on $k$, $l$, $|\Omega|$,
 $\|u_0\|_{L^\infty(\Omega)}$ and $\|w_0\|_{W^{1,\infty}(\Omega)}$ such that
\begin{align*}
&\alpha k(k-1)\int_\Omega u^{k-1}|\nabla w|^2 dx
 +C_5\int_\Omega u^2|\nabla w|^{2(l-1)}dx\\
&\le\frac{\alpha k(k-1)}{2(k+1)^2}\int_\Omega 
 \big|\nabla u^{\frac{k+1}2}\big|^2dx+\frac{\nu (l-1)}l
 \int_\Omega|\nabla |\nabla w|^l|^2dx+C_9\quad \text{for all } t\in (0,T^*).
\end{align*}
Combing this with \eqref{G3}, and setting 
$y_\delta(t):=\int_\Omega |\nabla w|^{2l}dx+\int_\Omega w^k dx$,
we conclude that
$y_\delta'(t)+y_\delta(t)\le C_9$ for all $ t\in(0,T^*)$.
Thus an ODE comparison argument  yields the uniform  boundedness of 
$y_\delta(t)$ on $(0,T^*)$, which implies that  
$\|u(\cdot,t)\|_{L^k(\Omega)}$ and $\|\nabla w(\cdot,t)\|_{L^{2l}(\Omega)}$ 
are uniformly bounded on $(0,T^*)$. This completes the proof of Lemma \ref{LkE}.  
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
 By Lemma \ref{LE}, there exists a unique local-in-time classical solution 
$(u,w)$ to equations \eqref{AH} on $[0,T^*)$. By \cite[Lemma A.1]{TW}  
and Lemma \ref{LkE},  we can establish  the uniform boundedness of $u$ 
and $\nabla w$ in $\Omega\times(0,T^*)$.
Then we can deduce that $T^*=\infty$  by using the extension criterion 
in Lemma \ref{LE}. Hence we have completed the proof of Theorem \ref{thm1} 
under the condition that 
$(u_0,w_0)\in C^2(\overline{\Omega})\times C^2(\overline{\Omega})$.  
For the case that $u_0$ is merely H\"{o}lder continuous and nonnegative 
in $\overline{\Omega}$ and that $v_0$ belongs to $W^{1,\infty}(\Omega)$ 
only, we can follow the corresponding proof in \cite{LTW} to conclude 
the proof.   
\end{proof}

\subsection*{Acknowledgements}  
The author is very grateful to the anonymous referees for their  
comments and valuable suggestions, which greatly improved this article. 
She also thanks the helpful discussions with Professor Yuan Lou.  
This work  was partially supported by NNSF of China (no. 11101068),  
Sichuan Youth Science \& Technology Foundation (no. 2011JQ0003) 
and by SRF for ROCS, SEM.

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\end{document}