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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 141, 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  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/141\hfil Existence of global solutions]
{Existence of global solutions for a system of
reaction-diffusion equations having a triangular matrix}

\author[E. H. Daddiouaissa\hfil EJDE-2008/141\hfilneg]
{El Hachemi Daddiouaissa}

\address{El Hachemi Daddiouaissa \newline
Department of Mathematics
University Kasdi Merbah, UKM
Ouargla 30000, Algeria}
\email{dmhbsdj@gmail.com}

\thanks{Submitted August 19, 2007. Published October 16, 2008.}
\subjclass[2000]{35K57, 35K45}
\keywords{Reaction-diffusion systems; Lyapunov functional; global solution}

\begin{abstract}
 We consider the system of reaction-diffusion equations
 \begin{gather*}
  u_{t}-a\Delta u=\beta-f(u,v)-\alpha u,\\
  v_{t}-c\Delta u-d\Delta v=g(u,v)-\sigma v.
  \end{gather*}
 Our aim is to establish the existence of global classical solutions
 using the method used by Melkemi, Mokrane, and Youkana \cite{melk}.
\end{abstract}

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

\section{Introduction}

In this manuscript, we consider a reaction-diffusion system that
arises in the study of physical, chemistry, and various biological
processes including population dynamics
\cite{cast,cuss,fitz,hama,kira,webb,zeid}.
The system of equations is
\begin{gather} \label{e1.1}
\frac{\partial u}{\partial t}-a\Delta u=\beta-f(u,v)-\alpha u
\quad (x,t)\in\Omega \times R_{+}\\
\frac{\partial v}{\partial t}-c\Delta u-d\Delta v=g(u,v)-\sigma v\quad
(x,t)\in\Omega \times R_{+}, \label{e1.2}
\end{gather}
with the boundary conditions
\begin{equation} \label{e1.3}
\frac{\partial u}{\partial \eta}=\frac{\partial v}{\partial
\eta}=0\quad\text{on } \partial \Omega \times R_{+}.
\end{equation}
and the initial data
\begin{equation} \label{e1.4}
0\leq u(0,x)=u_{0}(x) ;\quad 0\leq v(0,x)=v_{0}(x)\quad\text{in } \Omega.
\end{equation}
  where $\Omega$ is a smooth open bounded domain in $R^{n}$, with boundary
$\partial \Omega$ of class $C^{1}$ and $\eta$ is the outer normal to $\partial \Omega$.
 The constants of diffusion $a,c,d$ are such that
$a>0$, $d>0$, $c>0$ and $a>d$, $c^{2}<4ad$ which is the parabolic condition,
and $\alpha,\sigma $ are positive constants, $\beta\geq 0$, and $f,g$
are nonnegative functions of class $C(R_{+}\times R_{+})$, such that
\begin{itemize}
\item[(H1)] For all $\tau\geq 0$, $f(0,\tau)=0$;
\item[(H2)] For all $\xi\geq 0$ and all $\tau\geq 0$,
$0\leq f(\xi,\tau)\leq \varphi(\xi)(\mu+\tau)^{r}$;
\item[(H3)] For all $\xi\geq 0$ and all $\tau \geq 0$,
 $g(\xi,\tau)\leq \psi(\tau)f(\xi,\tau)+\phi(\tau)$,
\end{itemize}
where $r,\mu$ are positive constants such that $r\geq 1$,
$\mu\geq 1$, $\varphi,\psi$ and $\phi$ are nonnegative functions
of class $C(R^{+})$, such that
\begin{gather} \label{e1.5}
\lim_{\tau\to +\infty}\frac{\psi(\tau)}{\tau}
=\lim_{\tau\to +\infty}\frac{\phi(\tau)}{\tau}=0.\\
\phi(0)> \beta \frac{c}{a-d}. \label{e1.6}
\end{gather}
In addition we suppose that
\begin{equation} \label{e1.7}
g(\xi,\frac{c}{a-d}\xi)+\frac{c}{a-d}f(\xi,\frac{c}{a-d}\xi)
\geq\frac{c}{a-d}[(\sigma-\alpha)\xi+\beta],\quad \forall \xi\geq0.
\end{equation}
Melkemi et al \cite{melk} established the existence of global solutions,
(eventually uniformly bounded in time) using a novel approach that
involved the use of a Lyapunov function for system \eqref{e1.1}--\eqref{e1.4}
when $c=0$.
Here, we apply the same method to the study of system
\eqref{e1.1}--\eqref{e1.4} when $c>0$; that is, for
 a model that involves a triangular matrix.

\section{Existence of  local and positive solutions}

First we convert system \eqref{e1.1}--\eqref{e1.4}
into an abstract first order system in
$X=C(\overline{\Omega})\times C(\overline{\Omega})$ of the form
\begin{gather} \label{e2.1}
U'(t)=AU(t)+F(U(t)), \quad t>0,\\
U(0)=U_{0}\in X, \label{e2.2}
\end{gather}
where
\begin{gather*}
AU(t)=( a\Delta u, c\Delta u+d\Delta v), \\
F(U(t))=(\beta-f(u,v)-\alpha u,g(u,v)-\sigma v ).
\end{gather*}
Since $F$ is locally Lipschitz in $U$ and $X$,  for every initial
data $U_{0}\in X$, system \eqref{e2.1}--\eqref{e2.2}
admits a unique strong local solution on $]0,T^{*}[$, where $T^{*}$
is the eventual blowing-up time,
(see  Kirane \cite{kira},  Friedman \cite{frie},  Henry \cite{henr},
 Pazy \cite{pazy}).

Multiplying  \eqref{e1.1} by $\frac{c}{a-d}$, and subtracting the
resulting equation from \eqref{e1.2} leads to the system
\begin{gather} \label{e2.3}
\frac{\partial u}{\partial t}-a\Delta u=\Lambda(u,z), \quad
 (x,t)\in\Omega \times R_{+}\\
\frac{\partial z}{\partial t}-d\Delta z=\Upsilon(u,z),\quad
 (x,t)\in\Omega \times R_{+}. \label{e2.4}
\end{gather}
where
\begin{gather*}
\Lambda(u,z)=\beta-f(u,v)-\alpha u\\
\Upsilon(u,z)=g(u,v)+\frac{c}{a-d}f(u,v)
+\frac{c}{a-d}(\alpha u-\beta)-\sigma v, \\
z=v-\frac{c}{a-d}u,
\end{gather*}
with the boundary conditions
\begin{equation} \label{e2.5}
\frac{\partial u}{\partial \eta}=\frac{\partial z}{\partial
\eta}=0,\quad \text{on } \partial \Omega \times R_{+}
\end{equation}
and initial data
\begin{gather} \label{e2.6}
u(0,x)=u_{0}(x)\quad \text{in } \Omega,\\
z(0,x)=z_{0}(x)=v_{0}(x)-\frac{c}{a-d} u_{0}(x)\quad \text{in }\Omega. \label{e2.7}
\end{gather}
If we assume \eqref{e1.7} and (H1) then a simple application of
 a comparison theorem to system \eqref{e2.3}-\eqref{e2.4} implies
 (see \cite{kira}) that for positive initial data $u_{0}\geq0$ and
$z_{0}\geq o$ we have that
\[
u(t,x)\geq0, \quad v(t,x)\geq\frac{c}{a-d}u(t,x)\quad
 \forall (x,t)\in \Omega \times ]0,T^{*}[.
\]

\section{Existence of global solutions}

Before we establish the existence of a global solution, we introduce
some notation. Here, we let
\begin{equation*}
\| u\|_{p}^{p}=\frac{1}{|\Omega|}\int_{\Omega} |u(x)|^{p} dx
\text{ and } \| u\|_{\infty}= \max_{x\in\Omega}|u(x)|.
\end{equation*}
denote the usual norms in spaces $L^{p}(\Omega)$, $L^{\infty}(\Omega)$ 
and $C(\overline{\Omega})$.
Applying the comparison principle we get that
\begin{equation} \label{e3.1}
u(t,x)\leq \max(\| u_{0}\|_{\infty},\frac{\beta}{\alpha})=K.
\end{equation}
To establish the uniform boundedness of $v$, it is sufficient
 to show the uniform boundedness of $z$. This task is carried out using
a result found in Henry \cite[pp. 35-62]{henr}, from which it
sufficient to derive a uniform estimate for $\| \Upsilon(u,z) \|_{p}$;
 that is, finding a constant $C$ such that
\begin{equation} \label{e3.2}
\| \Upsilon(u,z) \|_{p}\leq C.
\end{equation}
Here C is a nonnegative constant independent of t, for some
$p>n/2$. The key is to  establish the uniform boundedness of
 $\| v\|_{p}$ on $]0,T^{*}[$, taking into account assumptions
(H2) and (H3).
When $p\geq 2$, we put
\begin{equation} \label{e3.3}
\begin{gathered}
\Gamma(p)=\frac{p\Gamma+1}{p-1}\,,\quad
\Gamma=(a-d)^{2}[1+\frac{1}{4ad}]\,,\\
l=\frac{2\beta \rho}{\Gamma(p)\sigma}\,,\quad
\omega= [\frac{S^{2}}{4adR^{2}}+\frac{p}{R^{2}}+\mu](p-1)
\end{gathered}
\end{equation}
where $\rho >0$,
$$
S=\frac{\rho}{\Gamma(p)l}, \quad
R=\frac{(a-d)\rho}{\Gamma(p)(l+\rho K)}.
$$
Using these definitions and notation we can state the following
key proposition

\begin{proposition} \label{prop3.1}
Assume that $p\geq 2$ and let
\begin{equation} \label{e3.4}
G_{N}(t)= N \int_{\Omega}u dx+\int_{\Omega}(v+\omega)^{p}
\exp(-\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho(K-u)]))dx,
\end{equation}
where $(u,v)$ is the solution of \eqref{e1.1}--\eqref{e1.4} on
$]0,T^{*}[$. Then under the assumptions (H3) and \eqref{e1.5}
 there exist two positive constants $N$ and $s$ such that
\begin{equation} \label{e3.5}
\frac{dG_{N}}{dt} \leq -(p-1)\sigma G_{N}+s.
\end{equation}
\end{proposition}

The proof of the above proposition requires some lemmas.

\begin{lemma} \label{lem3.1}
If $(u,v)$ be a solution of $\eqref{e1.1}-\eqref{e1.4}$ then
\begin{equation} \label{e3.6}
\int_{\Omega}f(u,v)dx\leq\beta|\Omega|-\frac{d}{dt}\int_{\Omega}u(t,x)dx.
\end{equation}
\end{lemma}

\begin{proof}
We integrate both sides of  \eqref{e1.1},
$$
f(u,v)=\beta -\alpha u -\frac{d}{dt}u(t,x)
$$
satisfied by $u$, which is positive and then we find \eqref{e3.6}.
\end{proof}

\begin{lemma} \label{lem3.2}
Assume that $p\geq2$, then under the assumptions {\rm (H1)--(H3)}, and
\eqref{e1.5} there exists $N_1$, such that
\begin{equation} \label{e3.7}
[p (g(\xi,\tau)-\phi(\tau))(\tau+\omega)^{p-1}
- \theta f(\xi,\tau) (\tau+\omega)^{p}]\leq  N_1f(\xi,\tau)
\end{equation}
for all $0\leq \xi \leq K$ and $\tau \geq 0$, $\theta > 0$.
\end{lemma}

\begin{proof}
from the assumption (H3) and \eqref{e1.5}, we conclude that there
exists $\tau_{0}>0$, such that for all $0\leq\xi\leq K,\tau\geq\tau_{0}$,
one finds
$$
[p\frac{\psi(\tau)}{\tau+\omega}-\theta ](\tau+\omega)^{p}f(\xi,\tau)\leq0
$$
now if $\tau$ is in the compact interval $[0,\tau_{0}]$, then the
continuous function
$$
\chi(\xi,\tau)=p\psi(\tau)(\tau+\omega)^{p-1}-\theta (\tau+\omega)^{p}
$$
is bounded.
\end{proof}

\begin{lemma} \label{lem3.3}
For all $\tau\geq 0$ and $\omega \geq 1$, we have
\begin{equation} \label{e3.8}
\frac{\beta \rho}{\Gamma(p)l} (\tau+\omega)^{p}
- \sigma p (\tau+\omega)^{p-1}\tau+p\phi(\tau)(\tau+\omega)^{p-1}\leq
 -(p-1)\sigma (\tau+\omega)^{p}+ M_1
\end{equation}
where $M_1$ is positive constant.
\end{lemma}

\begin{proof}
Let us put
\begin{align*}
\Pi=&\frac{\beta \rho}{\Gamma(p)l} (\tau+\omega)^{p}
  - \sigma p (\tau+\omega)^{p-1}\tau \\
\leq  &[2\frac{\beta \rho}{\Gamma(p)l}- \sigma p]
 \tau(\tau+\omega)^{p-1}+[2\frac{\beta \rho}{\Gamma(p)l}
 \frac{\omega}{\tau+\omega}-\frac{\beta \rho}{\Gamma(p)l}] (\tau+\omega)^{p}
\end{align*}
since $l=\frac{2\beta \rho}{\Gamma(p)\sigma}$, then
\begin{equation*}
\Pi+p\phi(\tau)(\tau+\omega)^{p-1}\leq -(p-1)\sigma (\tau+\omega)^{p}
+[\frac{p (\sigma \omega+\phi(\tau))}{\sigma(\tau+\omega)}
-\frac{1}{2}] \sigma (\tau+\omega)^{p}
\end{equation*}
using \eqref{e1.5} and Lemma \ref{lem3.2}, we can establish 
Proposition \ref{prop3.1}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop3.1}]
Let
$$
h(u)=-\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho(K-u)])
$$
so that
$$
G_{N}(t)= N \int_{\Omega}u dx+ L(t)
$$
where
$$
L(t)=\int_{\Omega}e^{h(u)} (v+\omega)^{p}dx.
$$
Differentiating $L$ with respect to $t$ and using Greens formula,
we obtain that
$$
L'(t)=I+J
$$
where
\begin{align*}
I=&- \int_{\Omega}[a(h''(u)+h'^{2}(u)) (v+\omega)^{p}+pc h'(u)
  (v+\omega)^{p-1}] e^{h(u)}\nabla u^{2} dx\\
&-\int_{\Omega} [p(a+d)h'(u) (v+\omega)^{p-1}+p(p-1)
 c (v+\omega)^{p-2}] e^{h(u)} \nabla u\nabla v dx\\
&-\int_{\Omega} p(p-1)d (v+\omega)^{p-2} e^{h(u)} \nabla v^{2} dx
\end{align*}
and
\begin{align*}
J=&\int_{\Omega}\beta h'(u)(v+\omega)^{p} e^{h(u)} dx\\
&+\int_{\Omega} [p g(u,v) (v+\omega)^{p-1}-h'(u) f(u,v) (v+\omega)^{p}]
  e^{h(u)}dx\\
&-\int_{\Omega} \alpha h'(u) u (v+\omega)^{p} e^{h(u)}dx
  - \int_{\Omega} \sigma p (v+\omega)^{p-1}v e^{h(u)}dx.
\end{align*}
We see that $I$ involves a quadratic form with respect to
$\nabla u,\nabla v$.
\begin{align*}
D=&[a(h''(u)+h'^{2}(u)) (v+\omega)^{p}+pch'(u)(v+\omega)^{p-1}] \nabla u^{2} \\
&+[p(a+d)h'(u)(v+\omega)^{p-1}+p(p-1)c(v+\omega)^{p-2}] \nabla u\nabla v \\
&+p(p-1)d(v+\omega)^{p-2} \nabla v^{2}\\
\end{align*}
which is nonnegative if
\begin{align*}
\delta=&[p(a+d)h'(u)(v+\omega)^{p-1}+p(p-1)(v+\omega)^{p-2}]^{2}\\
&-4p(p-1)d(v+\omega)^{p-2}[a(h''(u)+h'^{2}(u))(v+\omega)^{p}\\
&+pch'(u)(v+\omega)^{p-1}]\leq 0.
\end{align*}
Indeed,
\begin{align*}
\delta=&p^{2}(a+d)^{2}h'^{2}(u)(v+\omega)^{2p-2}+2(a+d)cp^{2}(p-1)h'(u))
  (v+\omega)^{2p-3}\\
&+c^{2}p^{2}(p-1)^{2}(v+\omega)^{2p-4}-4p(p-1)ad(h''(u)+h'^{2}(u))
  (v+\omega)^{2p-2}\\
&-4cdp^{2}(p-1)h'(u)(v+\omega)^{2p-3}
\end{align*}
and from \eqref{e3.3} we have that $v+\omega \geq 1$.
It follows that
\begin{align*}
\delta \leq &[p^{2}(a+d)^{2}h'^{2}(u)-4p(p-1)ad (h''(u)+h'^{2}(u))] (v+\omega)^{2p-2}\\
&+[2(a+d)c p^{2}(p-1)h'(u) +c^{2} p^{2}(p-1)^{2}\\
&-4cd p^{2}(p-1) h'(u)] (v+\omega)^{2p-3}.
\end{align*}
Let us try
$$
T=p^{2}(a+d)^{2}h'^{2}(u)-4p(p-1)ad (h''(u)+h'^{2}(u))
$$
the choice of $h(u)$ implies
\begin{equation} \label{e3.9}
\frac{\rho}{\Gamma(p)[l+\rho K]}\leq h'(u)\leq\frac{\rho}{\Gamma(p)l},
\end{equation}
and, consequently,
$$
T=-\frac{\rho^{2}4ad(a-d)^{2}p^{2}}{[\Gamma(p)(l+\rho (K-u))]^{2}}
 \leq -4adp^{2} R^{2}\leq 0.
$$
In addition
\begin{align*}
\delta\leq &[2(a+d)c p^{2}(p-1)h'(u)+c^{2} p^{2}(p-1)^{2}
 -4cd p^{2}(p-1) h'(u)] (v+\omega)^{2p-3}\\
&+T (v+\omega)(v+\omega)^{2p-3}\\
\leq &[(p-1)c^{2}+2(a-d)c\frac{\rho}{\Gamma(p)l}
 -\frac{4ad R^{2}\omega}{(p-1)}] p^{2}(p-1)(v+\omega)^{2p-3}
+T v(v+\omega)^{2p-3}.
\end{align*}
If we replace $w$ by its value \eqref{e3.3} and  use the parabolic condition
$c^{2}-4ad < 0$, we deduce that
\begin{align*}
\delta \leq&[p(c^{2}-4ad)-(c-S)^{2}-4ad R^{2}\mu] p^{2}(p-1)(v+\omega)^{2p-3}\\
&+T v(v+\omega)^{2p-3}\leq 0,
\end{align*}
that is, $I \leq 0$.

We can control the second term by observing that
\begin{align*}
J\leq&\int_{\Omega}[\beta h'(u) (v+\omega)^{p}
  - \sigma p (v+\omega)^{p-1}v+p\phi(v)(v+\omega)^{p-1}]e^{h(u)} dx\\
&+\int_{\Omega} [p (g(u,v)-\phi(v)) (v+\omega)^{p-1}-h'(u) f(u,v)
 (v+\omega)^{p}] e^{h(u)}dx.
\end{align*}
Using \eqref{e3.9},
\begin{align*}
J\leq&\int_{\Omega}[\frac{\beta \rho}{\Gamma(p)l} (v+\omega)^{p}
  - \sigma p (v+\omega)^{p-1}v+p\phi(v)(v+\omega)^{p-1}]e^{h(u)} dx\\
&+\int_{\Omega} [p (g(u,v)-\phi(v)) (v+\omega)^{p-1}
  - \frac{\rho}{\Gamma(p)[l+\rho k]} f(u,v) (v+\omega)^{p}] e^{h(u)}dx.
\end{align*}
Applying Lemmas \ref{lem3.2} and \ref{lem3.3}, one finds that
\[
J\leq-(p-1)\sigma\int_{\Omega}(v+\omega)^{p}e^{h(u)} dx
 +M_1\int_{\Omega}e^{h(u)} dx+N_1\int_{\Omega}f(u,v)e^{h(u)} dx.
\]
In addition we see that
$$
h(u)\leq -\frac{1}{\Gamma(p)}\ln\frac{2\beta\rho}{\sigma},
$$
and, consequently,
\[
J\leq -(p-1)\sigma L +M_1|\Omega|e^{-\frac{1}{\Gamma(p)}
\ln\frac{2\beta\rho}{\sigma}}+N_1e^{-\frac{1}{\Gamma(p)}
\ln\frac{2\beta\rho}{\sigma}}\int_{\Omega}f(u,v)dx.
\]
Letting
$$
M=M_1|\Omega|e^{-\frac{1}{\Gamma(p)}\ln\frac{2\beta\rho}{\sigma}},
\quad  N=N_1e^{-\frac{1}{\Gamma(p)}\ln\frac{2\beta\rho}{\sigma}}
$$
and using Lemma \ref{lem3.1} we conclude that
\begin{align*}
J\leq&-(p-1)\sigma L +M+N[\beta |\Omega|-\frac{d}{dt}
  \int_{\Omega} u(t,x) dx]\\
\leq &-(p-1)\sigma G_{N}+(p-1)\sigma N\int_{\Omega} u dx+ M
  +N \beta |\Omega|- N\frac{d}{dt}\int_{\Omega} u(t,x) dx\\
\leq &-(p-1)\sigma G_{N}+|\Omega|N[(p-1)K\sigma + \beta] + M
  - N\frac{d}{dt}\int_{\Omega} u(t,x) dx.
\end{align*}
It follows that
$$
\frac{dG_{N}}{dt} \leq -(p-1)\sigma G_{N}+s
$$
where $s=|\Omega|N[(p-1)K\sigma + \beta] + M$.
\end{proof}

We can now establish the main result of this manuscript.

\begin{theorem} \label{thm3.1}
Under the assumptions {\rm (H1)-(H3)} and \eqref{e1.5}, the solutions
of \eqref{e1.1}--\eqref{e1.4} are global and uniformly bounded
in $[0,+\infty[$.
\end{theorem}

\begin{proof}
Multiplying  \eqref{e3.5} by $e^{(p-1)\sigma t}$ and integrating,
implies the existence of a positive constant $C>0$ independent of $t$
such that
$$
G_{N}(t)\leq C.
$$
Since
$$
e^{h(u)}\geq e^{-\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho k)])},
$$
 for all $p\geq 2$, we have
\[
\int_{\Omega}(v+\omega)^{p} dx\quad
\leq e^{\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho k))}G_{N}(t)
\leq C e^{\frac{1}{\Gamma(p)}\ln \Gamma(p)[l+\rho k]}= C(p).
\]
Consequently,
$$
\int_{\Omega}(v+\mu)^{p} dx \leq C(p),\quad
\int_{\Omega}v^{p} dx \leq C(p).
$$
Now we chose $p> n/2$ and we search for a bound for
$\| \Upsilon(u,v)\|_{p}$.
We put
$$
A_1= \max_{0\leq \tau\leq \tau_{0}} \psi(\tau),\quad
A_2=\max_{0\leq \xi\leq K}\varphi(\xi),\quad
A_3 = \max_{0\leq \tau\leq \tau_{0}} \phi(\tau)
$$
where $\tau_{0}=max(\tau_1,\tau_2)$ such that
$$
\tau\geq \tau_1 \quad \Rightarrow \quad \psi(\tau)\leq \tau, \quad
\tau\geq \tau_2 \quad \Rightarrow \quad  \phi(\tau)\leq \tau.
$$
Using {\rm (H1)--(H3)} implies
$$
g(u,v)\leq\psi(v) f(u,v)+\phi(v)
\leq \psi(v) \varphi(u) (\mu+v)^{r}+\phi(v)
\leq A_2 \psi(v) (\mu+v)^{r}+\phi(v).
$$
Since $0\leq u\leq K$, we have
\begin{equation*}
\int_{\Omega} g(u,v)^{p} dx\leq \int_{\Omega} [A_2 \psi(v) (\mu+v)^{r}
 +\phi(v)]^{p}dx.
\end{equation*}
Now we use the inequality
$$
(x+y)^{q}\leq 2^{q-1}(x^{q}+y^{q})
$$
all $x,y\geq0$ and $q\geq1$, to obtain the following sequence of estimates
\begin{align*}
&\int_{\Omega} g(u,v)^{p} dx\\
&\leq \int_{\Omega}2^{p-1}[A_2^{p} \psi(v)^{p} (\mu+v)^{rp}+\phi(v)^{p}]dx\\
&\leq  2^{p-1} \Big[A_2^{p}\Big( \int_{v\leq \tau_{0}} A_1^{p}
 (\mu+\tau_{0})^{rp} dx+\int_{v\geq \tau_{0}} v^{p} (\mu+v)^{rp} dx\Big)
 + |\Omega| A_3^{p}+\int_{v\geq \tau_{0}} v^{p}dx\Big] \\
&\leq  2^{p-1} \Big[\Big(A_2 A_1(\mu+\tau_{0})^{r}\Big)^{p}|\Omega|
 + A_2^{p}\int_{v\geq \tau_{0}} (\mu+v)^{(r+1)p} dx
 + |\Omega| A_3^{p}+\int_{v\geq \tau_{0}} v^{p}dx\Big]\\
&\leq  2^{p-1} [(A_2 A_1(\mu+\tau_{0})^{r})^{p}|\Omega|+ A_2^{p}C((r+1)p)
+ |\Omega| A_3^{p}+C(p)]
=E_{g}^{p}\\
\end{align*}
and
\[
\int_{\Omega} f(u,v)^{p} dx\leq  A_2^{p} C(rp)=E_{f}^{p}.
\]
We conclude that
\begin{align*}
\|\Upsilon(u,v)\|_{p}
& \leq \|g(u,v)\|_{p}  +\frac{c}{a-d}[\|f(u,v)\|_{p}+\alpha\|u\|_{p}
 +\beta |\Omega|] +\sigma \|v\|_{p} \\
&\leq  E_{g}+\frac{c}{a-d}[E_{f}+(\alpha K+\beta)|\Omega|]
+\sigma \sqrt[p]{C(p)}.
\end{align*}
We conclude that the unique solution of \eqref{e1.1}--\eqref{e1.4}
 is globally and uniformly bounded in $[0,+\infty[\times\Omega$.
\end{proof}

\subsection*{Acknowledgments}
The author wants to thank Professors A. Youkana and M. Mechab for
their kind advice, and the anonymous referee for his/her comments that
improved the final version of this article.

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