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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 209, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/209\hfil Boundedness in a chemotaxis system]
{Boundedness in a chemotaxis system with consumption of
  chemoattractant and \\ logistic source}

\author[L. Wang, S. U.-D. Khan, S. U.-D. Khan \hfil EJDE-2013/209\hfilneg]
{Liangchen Wang, Shahab Ud-Din Khan, Salah Ud-Din Khan} 

\address{Liangchen Wang \newline
College of Mathematics and Statistics,
Chongqing University, Chongqing 401331, China}
\email{liangchenwang324@126.com}

\address{Shahab Ud-Din Khan  \newline
College of Mathematics and Statistics,
Chongqing University, Chongqing 401331, China}
\email{sudkhan@163.com}

\address{Salah Ud-Din Khan \newline
College of Engineering, King Saud University, P.O. Box 800,
Riyadh 11421, Kingdom of Saudi Arabia}
\email{salahudkhan@126.com}

\thanks{Submitted July 1, 2013. Published September 19, 2013.}
\subjclass[2000]{35B35, 35K55, 92C17}
\keywords{Chemotaxis; global existence; boundedness; logistic source}

\begin{abstract}
In this article, we consider a chemotaxis system with consumption
 of chemoattractant and logistic source
 \begin{gather*}
 u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+f(u),\quad x\in \Omega,\; t>0,\\
 v_t=\Delta v-uv,\quad  x\in\Omega,\; t>0,
 \end{gather*}
 under homogeneous Neumann boundary conditions in a smooth bounded domain
 $\Omega\subset \mathbb{R}^n$, with non-negative initial data $u_0$ and
 $v_0$ satisfying $(u_0,v_0)\in (W^{1,\theta}{(\Omega)})^2$
 (for some $\theta>n$). $\chi>0$ is a parameter referred to as
 chemosensitivity and $f(s)$ is assumed to generalize the logistic function
 \[
 f(s)=as-bs^2,\quad  s\geq0,\text{ with } a>0,\;b>0.
 \]
 It is proved that if $\|v_0\|_{L^\infty(\Omega)}>0$ is sufficiently small
 then the corresponding initial-boundary value problem possesses a unique
 global classical solution that is uniformly bounded.
\end{abstract}

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


\section{Introduction}

 This article considers the following chemotaxis system with consumption
of chemoattractant and logistic source
\begin{equation} \label{e1.1}
\begin{gathered}
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+f(u),\quad x\in \Omega,\; t>0,\\
v_t=\Delta v-uv,\quad x\in \Omega,\;  t>0,\\
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0,\quad
x\in \partial\Omega,\; t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \Omega,
\end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^n$ is a bounded domain with smooth
boundary $\partial\Omega$, and $\partial/\partial\nu$ denotes the
derivative with respect to the outer normal of $\partial\Omega$.
The parameter $\chi>0$ is referred as chemosensitivity, and the
function $f\in C^1([0,\infty))$ with $f(0)=0$. Moreover, we shall
suppose that
\begin{equation} \label{e1.2}
f(u)\leq au-bu^p\quad \text{for all } u\geq0
\end{equation}
with some $a>0$, $b>0$ and $p>1$.

Equations \eqref{e1.1} is the well-known Keller-Segel model, and the
origin of this fundamental model was introduced by Keller and Segel \cite{1}
to describe the motion of cells which are diffusing and moving towards
the concentration gradient of a chemical signal substance called
chemoattractant, the latter being produced by the cells themselves.
We refer the reader to the paper \cite{2} where a comprehensive
information of further examples illustrating the outstanding biological
relevance of chemotaxis can be found. In this paper, we consider
a mathematical model for the motion of cells which towards the higher
concentration of oxygen that is consumed by the cells, where $u=u(x, t)$
denotes the density of the cells and $v=v(x, t)$ represents the concentration
of the oxygen.

During the past four decades, the Keller-Segel models have become one of
the best-study models in mathematical biology the Keller-Segel models
have been studied extensively by many authors. For example, Keller
and Segel \cite{1} proposed the following classical chemotaxis model
\begin{equation} \label{e1.3}
\begin{gathered}
u_t=\Delta u-\nabla\cdot(u\nabla v),\quad x\in \Omega,\; t>0,\\
v_t=\Delta v-v+u,\quad x\in\Omega,\; t>0,
\end{gathered}
\end{equation}
which has been investigated successfully up to now and the main issue
of the investigation was the solutions of the model are bounded or blow-up.
If $n=1$, in \cite{3}, it was shown all solutions of \eqref{e1.3} are
global in time and bounded; if $n=2$, then all solutions of \eqref{e1.3}
are global in time and bounded provided that $\|u_0\|_{L^1{(\Omega)}}<4\pi$
in \cite{4}, however, for almost every $\|u_0\|_{L^1{(\Omega)}}>4\pi$,
then the corresponding solutions of \eqref{e1.3} blow up either in finite
or infinite time in \cite{5} and that some radially symmetric solutions
blow up in finite time in \cite{6,7}; if $n\geq3$,
Winkler \cite{8} showed that $\|u_0\|_{L^{n/2+\epsilon}{(\Omega)}}$
and $\|\nabla v_0\|_{L^{n+\epsilon}{(\Omega)}}$ are small for all
$\epsilon>0$, then the solution is global in time and bounded, however,
for any $\|u_0\|_{L^1{(\Omega)}}>0$, then the radially symmetric
solution of \eqref{e1.3} blows up either in finite or infinite time
(see also \cite{7,9,10}).

Involving a source term of logistic type in chemotaxis system have
been studied \cite{2,11,12,13,14,h2}. The following initial-boundary
value chemotaxis model with logistic source
\begin{equation} \label{e1.4}
\begin{gathered}
u_t=\Delta u-\nabla\cdot(u\chi(v)\nabla v)+f(u),\quad x\in \Omega,\;  t>0,\\
v_t=\Delta v-v+u,\quad x\in\Omega,\;  t>0.
\end{gathered}
\end{equation}
If $\chi(v)$ is a constant, Winkler \cite{11} studied  proved that the
solutions of problem \eqref{e1.4} are global and bounded provided
that $f(0)\geq0$ as well as $f(u)\leq a-bu^2$ with some $a\geq0$ and $b$
is sufficiently large. If
$\chi(v)\leq\frac{\chi_0}{(1+\beta v)^\delta}$ for all $v\geq0$
and some $\delta>1$, $\chi_0>0$ and  $\beta>0$, the authors \cite{h2}
shown that the model \eqref{e1.4} with logistic source $f(u)$
satisfies \eqref{e1.2} with $p=2$ then solutions are global and bounded
provided that $\chi_0$ and $a$ are sufficiently small,
the authors \cite{hh} recent obtain the same result for all positive
values of $\chi_0$ and $a$, which improved the previous result.

The model \eqref{e1.1} deals with the chemotaxis process where the
signal is consumed by the cells, rather than produced by the cells.
In the absence of the logistic source (i.e. $f(u)\equiv0$) for
problem \eqref{e1.1}, Tao \cite {18} proved that the classical solution
 of model \eqref{e1.1} is uniformly bounded provided that
$\|v_0\|_{L^\infty(\Omega)}$ is sufficiently small.
In particular, if $\Omega\subset \mathbb{R}^3$ is a bounded convex domains,
Tao and Winkler \cite {27} showed that there exists $T>0$ such that
the problem has global weak solution which is bounded and smooth in
$\Omega\times(T,+\infty)$. It is the goal of this paper to prove
that model \eqref{e1.1} has global and bounded solutions provided that
$\|v_0\|_{L^\infty(\Omega)}>0$ is sufficiently small (Theorem \ref{thm3.2}).


\section{Preliminaries}

We first state one result concerning local-in-time existence of a
classical solution to problem \eqref{e1.1}.

\begin{theorem} \label{thm2.1}
 Let the non-negative functions $u_0$ and $v_0$ satisfy
$(u_0,v_0)$ belong to $(W^{1,\theta}{(\Omega)})^2$,
for some $\theta>n$.
Moreover, $f(s)$ with $s\geq0$ is smooth and $f(0)=0$.
Then problem \eqref{e1.1} has a unique local-in-time non-negative
classical solution
\begin{equation} \label{e2.1}
(u, v)\in (C([0,T_{\rm max}); W^{1,\theta}{(\Omega)})
\cap C^{2,1}(\overline{\Omega}\times(0,T_{\rm max})))^2,
\end{equation}
where $T_{\rm max}$ denotes the maximal existence time.
If for every $T<+\infty$ both $u(\cdot,t)$ and $v(\cdot,t)$ are a
priori bounded for all $0<t<\min \{T,T_{\rm max}\}$; i.e.,
\begin{equation} \label{e2.2}
\|(u(\cdot,t),v(\cdot,t))\|_{L^{\infty}(\Omega)}\leq C(T)
\quad \text{for all } 0<t<\min \{T,T_{\rm max}\}
\end{equation}
with some constant $C(T)$ depending on $T$ and
$\|(u_0,v_0)\|_{W^{1,\theta}(\Omega)}$ only. Then the solution
of \eqref{e1.1} is globally defined and thus $T_{\rm max}=+\infty.$
Moreover, assume the condition \eqref{e1.2} is satisfied,
then there exists a constant $C_0>0$ such that $u(x,t)$ and $v(x,t)$ satisfy
\begin{gather} \label{e2.3}
\|u(\cdot,t)\|_{L^1(\Omega)}\leq C_0, \\
 \label{e2.4}
0\leq v\leq \|v_0\|_{L^\infty(\Omega)}
\end{gather}
for all $t\in(0,T_{\rm max})$.
\end{theorem}

\begin{proof}
 As in \cite{21,18,24}, let $V=(u,v)\in \mathbb{R}^2$. Then the
initial-boundary value problem \eqref{e1.1} can be reformulated as
\begin{gather*}
V_t=\nabla\cdot(F(V)\nabla V)+H(V)\\
\frac{\partial V}{\partial \nu}=0,\quad x\in \partial\Omega,\; t>0,\\
V(x,0)=(u_0(x),v_0(x)),\quad x\in \Omega,
\end{gather*}
where
\[
F(V)=  \begin{pmatrix}
1 & -\chi u \\
0 & 1  \\
\end{pmatrix}, \quad
H(V)=\begin{pmatrix} f(u) \\ -uv\end{pmatrix}.
\]
Then applying \cite[Theorems 14.4, 14.6, 15.5]{d4},
statements \eqref{e2.1} and \eqref{e2.2} can be proved.
Since the initial data $u_0\geq0$, $v_0\geq0$ and $f(0)=0$,
the maximum principle ensures that both $u$ and $v$ are non-negative.

By the maximum principle we have \eqref{e2.4}.
Now, we prove \eqref{e2.3}. Integrating the first equation in \eqref{e1.1}
and using \eqref{e1.2}, we obtain
\begin{equation} \label{e2.5}
\frac{d}{dt}\int_{\Omega}udx=\int_{\Omega}f(u)dx\leq\int_{\Omega}au-bu^pdx.
\end{equation}
By Young's inequality, since $b>0$ and $p>1$, we obtain
\begin{equation} \label{e2.6}
(a+1)\int_{\Omega}u\,dx
=\int_{\Omega}b^{-1/p}(a+1)b^{1/p}u\,dx
\leq \int_{\Omega}bu^pdx+(a+1)^\frac{p}{p-1}b^\frac{1}{1-p}|\Omega|.
\end{equation}
Combining \eqref{e2.5} and \eqref{e2.6}, we conclude
\begin{equation} \label{e2.7}
\frac{d}{dt}\int_{\Omega}udx+\int_{\Omega}u\,dx
\leq (a+1)^\frac{p}{p-1}b^\frac{1}{1-p}|\Omega|.
\end{equation}
Integrating, we have
\[
\int_{\Omega}udx\leq c_0, \quad
c_0=\max \{\|u_0\|_{L^1(\Omega)}, (a+1)^\frac{p}{p-1}b^\frac{1}{1-p}|\Omega|\}>0.
\]
\end{proof}
Let us collect some basic statements about the Gagliardo-Nirenberg
inequality which will be used in forthcoming proofs. For details,
we refer the reader to \cite{28,29,30} (see also \cite{9,10}).\

\begin{lemma} \label{lem2.2}
 Let
\[
\alpha^*=\begin{cases}
\frac{2n}{n-2}, &\text{if }n>2,\\
\infty,     &\text{if }n=1,2.
\end{cases}
\]
Then for all $l^*\in(2,\infty)$ satisfying $l^*\leq \alpha^*$
and $h\in(0,2)$, $\alpha\in[h,l^*]$, there exists a constant $c_{GN}>0$
such that
\[
\|\psi\|_{L^\alpha(\Omega)}\leq c_{GN}(\|\nabla \psi\|_{L^2(\Omega)}^{\lambda^*}
\|\psi\|_{L^h(\Omega)}^{1-\lambda^*}+\|\psi\|_{L^h(\Omega)})
\]
holds for any $\psi\in W^{1,2}(\Omega)$, where
$\lambda^*=\frac{\frac{n}{h}-\frac{n}{\alpha}}{1-\frac{n}{2}+\frac{n}{h}}$.
\end{lemma}

\section{Global bounded solutions}

The main step towards the  existence and boundedness of a global  solution
is to establish uniform bound of the cells population density $u(x,t)$ in
the space $L^{n+1}(\Omega)$. This is accomplished by providing some
associated weighted bounds involving weight functions $\phi(v)$
which are uniformly bounded both from above and below by positive constants.
This approach was developed by Winkler in \cite{16} (see also \cite{h2,18}).

\begin{lemma} \label{lem3.1}
 Let $f(u)$ satisfy \eqref{e1.2}, $\|v_0\|_{L^\infty(\Omega)}>0$ and $\chi>0$.
Then there exists a constant $C>0$ such that the first component of
the solution of \eqref{e1.1} satisfies
\begin{equation} \label{e3.1}
\|u(\cdot,t)\|_{L^{n+1}(\Omega)}\leq C\quad \text{for all }
 t\in(0,T_{\rm max}).
\end{equation}
\end{lemma}

\begin{proof}
Set $k:=n+1$ and fix $\|v_0\|_{L^\infty(\Omega)}>0$ small such that
\begin{equation} \label{e3.2}
\|v_0\|_{L^\infty(\Omega)}\leq\frac{1}{6(n+1)\chi}.
\end{equation}
Define
$$
\phi(s):=e^{(\alpha s)^2}\quad \text{for all }
 0\leq s\leq \|v_0\|_{L^\infty(\Omega)},
$$
where
\[
\alpha=\sqrt{\frac{n}{24(n+1)}}\frac{1}{\|v_0\|_{L^\infty(\Omega)}}.
\]
By direct calculation, from \eqref{e1.1}, we obtain
\begin{align}
&\frac{1}{k}\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx \nonumber \\
&=\int_{\Omega}u^{k-1}\phi(v)u_t dx+\frac{1}{k}\int_{\Omega}u^k \phi'(v)v_t\,dx
 \nonumber\\
&=\int_{\Omega}u^{k-1}\phi(v)\Delta u dx
 -\int_{\Omega}u^{k-1}\phi(v)\chi\nabla\cdot(u\nabla v)dx
 +\int_{\Omega}u^{k-1}\phi(v)f(u)dx \nonumber \\
&\quad +\frac{1}{k}\int_{\Omega}u^k \phi'(v)\Delta v\,dx
 -\frac{1}{k}\int_{\Omega}u^{k+1}v \phi'(v)dx \nonumber \\
&=-(k-1)\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx
 -\int_{\Omega}u^{k-1} \phi'(v)\nabla u\cdot\nabla v\,dx \nonumber \\
&\quad+\chi(k-1)\int_{\Omega}u^{k-1}\phi(v)\nabla u\cdot\nabla v\,dx
 +\chi\int_{\Omega}u^k\phi'(v)|\nabla v|^2dx \nonumber \\
&\quad +\int_{\Omega}u^{k-1}\phi(v)f(u)dx-\int_{\Omega}u^{k-1}
\phi'(v)\nabla u\cdot\nabla v\,dx \nonumber \\
&\quad -\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx
 -\frac{1}{k}\int_{\Omega}u^{k+1}v \phi'(v)dx. \label{e3.3}
\end{align}
Since $f(s)\leq as-bs^p$ and $\phi'(s)\geq0$ for all $s\geq0$, we have
\begin{equation} \label{e3.4}
\begin{split}
&\frac{1}{k}\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
 +(k-1) \int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx
 +\frac{1}{k} \int_{\Omega}u^k \phi''(v)|\nabla v|^2dx\\
&\leq -2\int_{\Omega}u^{k-1} \phi'(v)\nabla u\cdot\nabla v\,dx
 +\chi(k-1)\int_{\Omega}u^{k-1}\phi(v)\nabla u\cdot\nabla v\,dx\\
&\quad +\chi\int_{\Omega}u^k\phi'(v)|\nabla v|^2dx
 +a\int_{\Omega}u^k\phi(v)dx-b\int_{\Omega}u^{k+p-1}\phi(v)dx.
\end{split}
\end{equation}
By Young's inequality, we obtain
\begin{equation} \label{e3.5}
\begin{split}
-2\int_{\Omega}u^{k-1} \phi'(v)\nabla u\cdot\nabla v\,dx 
&\leq\frac{k-1}{4}\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx\\
&+\frac{4}{k-1}\int_{\Omega}u^k\frac{\phi'^2(v)}{\phi(v)}|\nabla v|^2dx
\end{split}
\end{equation}
and
\begin{equation} \label{e3.6}
\begin{split}
\chi(k-1)\int_{\Omega}u^{k-1}\phi(v)\nabla u\cdot\nabla v\,dx
&\leq\frac{k-1}{4}\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx\\
& +\chi^2(k-1)\int_{\Omega}u^k\phi(v)|\nabla v|^2dx.
\end{split}
\end{equation}
Thus, from \eqref{e3.4}--\eqref{e3.6} we obtain
\begin{equation} \label{e3.7}
\begin{split}
&\frac{1}{k}\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
 +\frac{k-1}{2}\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx
 +\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx\\
&\leq \frac{4}{k-1}\int_{\Omega}u^k\frac{\phi'^2(v)}{\phi(v)}|\nabla v|^2dx
 +\chi^2(k-1)\int_{\Omega}u^k\phi(v)|\nabla v|^2dx\\
&\quad +\chi\int_{\Omega}u^k\phi'(v)|\nabla v|^2dx
 +a\int_{\Omega}u^k\phi(v)dx-b\int_{\Omega}u^{k+p-1}\phi(v)dx.
\end{split}
\end{equation}
Next we show that the three terms on the right-hand side of \eqref{e3.7} 
are dominated by $\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx$. 
To this end, for $s\geq0$, we compute
\begin{gather*}
y_1(s):=\frac{\phi''(s)}{k}=\frac{2}{k}\alpha^2e^{(\alpha s)^2}+\frac{4}{k}\alpha^4s^2e^{(\alpha s)^2},\\
y_2(s):=\frac{4}{k-1}\frac{\phi'^2(s)}{\phi(s)}=\frac{16}{k-1}\alpha^4s^2e^{(\alpha s)^2},\\
y_3(s):=\chi^2(k-1)\phi(s)=\chi^2(k-1)e^{(\alpha s)^2},\\
y_4(s):=\chi\phi'(s)=2\chi\alpha^2se^{(\alpha s)^2}.
\end{gather*}
By a direct calculation, we obtain
\begin{equation} \label{e3.8}
\frac{y_2(s)}{\frac{1}{3}y_1(s)}
\leq \frac{\frac{16}{k-1}\alpha^4s^2e^{(\alpha s)^2}}{\frac{2}{3k}
 \alpha^2e^{(\alpha s)^2}}
= \frac{24k}{k-1}(\alpha s)^2
\leq \frac{24(n+1)}{n}(\alpha \|v_0\|_{L^\infty(\Omega)})^2
=1,
\end{equation}
where we have used that 
$\alpha=\sqrt{\frac{n}{24(n+1)}}\frac{1}{\|v_0\|_{L^\infty(\Omega)}}$. 
Using \eqref{e3.2},
\begin{equation} \label{e3.9}
\frac{y_3(s)}{\frac{1}{3}y_1(s)}
\leq \frac{\chi^2(k-1)e^{(\alpha s)^2}}{\frac{2}{3k}\alpha^2e^{(\alpha s)^2}}
\leq \frac{3k(k-1)\chi^2}{2\alpha^2}
=36(n+1)^2\|v_0\|^2_{L^\infty(\Omega)}\chi^2
\leq 1
\end{equation}
and
\begin{equation} \label{e3.10}
\frac{y_4(s)}{\frac{1}{3}y_1(s)}
\leq \frac{2\chi\alpha^2se^{(\alpha s)^2}}{\frac{2}{3k}\alpha^2e^{(\alpha s)^2}}
\leq 3k\chi s
\leq 3(n+1)\chi\|v_0\|_{L^\infty(\Omega)}
\leq \frac{1}{2}.
\end{equation}
Therefore,  from \eqref{e3.7}-\eqref{e3.10}, it follows easily that
\begin{equation} \label{e3.11}
\begin{split}
&\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
 +kb\int_{\Omega}u^{k+p-1}\phi(v)dx
 +\frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\phi(v)dx\\
&\leq ka\int_{\Omega}u^k\phi(v)dx.
\end{split}
\end{equation}
Since $ 0\leq s\leq \|v_0\|_{L^\infty(\Omega)}$, we have 
$1\leq\phi(s)\leq e^{(\alpha\|v_0\|_{L^\infty(\Omega)})^2}:=d$, 
it is not difficult to obtain
\begin{equation} \label{e3.12}
kb\int_{\Omega}u^k\phi(v)dx\leq kb\int_{\Omega}u^{k+p-1}\phi(v)dx+kbd|\Omega|.
\end{equation}
Combining \eqref{e3.11} with \eqref{e3.12} yields
\begin{equation} \label{e3.13}
\begin{split}
&\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
 +kb\int_{\Omega}u^k\phi(v)dx
 +\frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\phi(v)dx\\
&\leq ka\int_{\Omega}u^k\phi(v)dx+kbd|\Omega|.
\end{split}
\end{equation}
Using Lemma \ref{lem2.2} and  $(x+y)^\gamma\leq2^\gamma(x^\gamma+y^\gamma)$ 
for all $x,y\geq0$ and $\gamma>0$, we obtain
\begin{equation} \label{e3.14}
\begin{split}
ka\int_{\Omega}u^k\phi(v)dx
&\leq kad\int_{\Omega}u^kdx\\
&=kad\|u^{k/2}\|_{L^2(\Omega)}^2\\
&\leq kad(c_{GN}\|\nabla u^{k/2}\|_{L^2(\Omega)}^\lambda
 \|u^{k/2}\|_{L^{2/k}(\Omega)}^{1-\lambda}
 +c_{GN}\|u^{k/2}\|_{L^{2/k}(\Omega)})^2\\
&\leq 4kad(c^2_{GN}c_0^{k(1-\lambda)}
 \|\nabla u^{k/2}\|_{L^2(\Omega)}^{2\lambda}+c^2_{GN}c_0^k)
\end{split}
\end{equation}
holds with some constant $c_{GN}>0$ and 
$$
\lambda=\frac{\frac{kn}{2}-\frac{n}{2}}{1-\frac{n}{2}+\frac{kn}{2}}\in(0,1).
$$
By Young's inequality, we derive
\begin{equation} \label{e3.15}
\begin{split}
ka\int_{\Omega}u^k\phi(v)dx
&\leq 4kadc^2_{GN}c_0^{k(1-\lambda)}\|\nabla u^{k/2}\|_{L^2(\Omega)}
 ^{2\lambda}+4kadc^2_{GN}c_0^k\\
&\leq \frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2dx+c_1\\
&\leq \frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\phi(v)dx+c_1,
\end{split}
\end{equation}
where 
\[
c_1=c_0^k(4kadc^2_{GN}(\frac{2k-2}{k})^{-\lambda})^{\frac{1}{1-\lambda}}
+4kadc^2_{GN}c_0^k>0.
\]
 Hence, substituting \eqref{e3.15} into \eqref{e3.13} yields
\begin{equation} \label{e3.16}
\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx+kb\int_{\Omega}u^k\phi(v)dx
\leq c_1+kbd|\Omega|.
\end{equation}
Integrating \eqref{e3.16}, we have
\begin{equation*}
\int_{\Omega}u^kdx\leq\int_{\Omega}u^k \phi(v)dx
\leq \max \big\{d\int_{\Omega}u_0^k, \frac{c_1+kbd|\Omega|}{kb}\big\},
\end{equation*}
we arrive at the desired result. 
\end{proof}

\begin{remark} \label{rmk1} \rm
To prove that the three terms on the right-hand side of \eqref{e3.7} 
are dominated by $\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx$,
we need  $\frac{y_i(s)}{\frac{1}{3}y_1(s)}\leq1$ $(i=2,3,4)$, so we have 
$0<\|v_0\|_{L^\infty(\Omega)}\leq\frac{1}{6k\chi}$, in such a way that 
$\frac{1}{6k\chi}\to 0$ as $k\to \infty$. 
In fact, in the proof of Theorem \ref{thm3.2} is only applied to one fixed $k>n$.
So  to avoid this  situation, we choose $k:=n+1$ in Lemma \ref{lem3.1}.
\end{remark}

We are now in a position to prove our main results,
which are as follows.

\begin{theorem} \label{thm3.2}
 Assume that $u_0(x)$ and $v_0(x)$ are non-negative functions and that
 $(u_0,v_0)$ belongs to $(W^{1,\theta}{(\Omega)})^2$  
for some $\theta>n$, $\chi>0$, $f(u)$ satisfies \eqref{e1.2}. 
Then problem \eqref{e1.1} possesses a unique global classical solution 
$(u,v)$ for which both $u$ and $v$ are non-negative and uniformly bounded 
in $\Omega\times(0,\infty)$ provided that
$$
0<\|v_0\|_{L^\infty(\Omega)}\leq\frac{1}{6(n+1)\chi}.
$$
\end{theorem}

\begin{proof} 
With the aid of Lemma \ref{lem3.1} and its proof, based on a Moser-Alikakos-type 
iterative procedure \cite{19,20}
 (for detailed calculations we refer to \cite{h2,18,26}),
 we can establish a uniform bound on the solution $u$ in time $(0,T_{\rm max})$. 
Combining \eqref{e2.4} and \eqref{e2.2} we obtain the desired result 
of Theorem \ref{thm3.2}.
\end{proof}

\subsection*{Acknowledgments}
We would like to thank the anonymous reviewers for their valuable suggestions 
and fruitful comments which led to significant improvement of this work. 
The third author would like to extend his sincere appreciation to 
deanship of scientific research at King Saud University for its 
funding of this research through the research group project no. RGP-VPP-255.

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\end{document}