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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 94, pp. 1--11.\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-2015/94\hfil Uniform Boundedness of solutions]
{Boundedness and large-time behavior of solutions for a
Gierer-Meinhardt system of three equations}

\author[S. Henine, S. Abdelmalek,  A. Youkana \hfil EJDE-2015/94\hfilneg]
{Safia Henine, Salem Abdelmalek, Amar Youkana}

\address{Safia Henine \newline
Department of Mathematics, University of Batna 05000, Algeria}
\email{henine.safia@yahoo.fr}

\address{Salem Abdelmalek \newline
Department of Mathematics, College of Sciences, Yanbu Taibah
University, Saudi Arabia. \newline
Department of Mathematics,
University of Tebessa 12002, Algeria}
\email{sabdelmalek@taibahu.edu.sa}

\address{Amar Youkana \newline
Department of Mathematics, University of Batna 05000, Algeria}
\email{youkana\_amar@yahoo.fr}

\thanks{Submitted January 8, 2015. Published April 14, 2015.}
\subjclass[2000]{35K57}
\keywords{Gierer-Meinhardt system; Lyapunov functional; \hfill\break\indent Uniform boundedness}

\begin{abstract}
 The aim of this work is to prove the uniform boundedness and the existence
 of global solutions for Gierer-Meinhardt  model of three substance described
 by reaction-diffusion equations with Neumann boundary conditions.
 Based on a Lyapunov functional we establish the asymptotic behaviour of
 the solutions.
\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}

In this article, we consider the Gierer-Meinhardt type system of three
equations
\begin{equation}\label{1.1}
\begin{gathered}
\frac{\partial u}{\partial t}-a_1\Delta u=-b_1u+f(u,v,w),\quad
 \text{in } \mathbb{R}^{+}\times\Omega,\\
\frac{\partial v}{\partial t}-a_2\Delta v=-b_2v+g(u,v,w),\quad
 \text{in } \mathbb{R}^{+}\times\Omega,\\
\frac{\partial w}{\partial t}-a_{3}\Delta w=-b_{3}w+h(u,v,w),\quad
 \text{in } \mathbb{R}^{+}\times\Omega,
\end{gathered}
\end{equation}
where
\begin{gather*}
f(u,v,w)=\rho_1(x,u,v,w)\frac{u^{p_1}}{v^{q_1}(w^{r_1}+c)}+\sigma_1(x),\\
g(u,v,w)=\rho_2(x,u,v,w)\frac{u^{p_2}}{v^{q_2}w^{r_2}}+\sigma_2(x),\\
h(u,v,w)=\rho_{3}(x,u,v,w)\frac{u^{p_{3}}}{v^{q_{3}}w^{r_{3}}}+\sigma_{3}(x),
\end{gather*}
with homogeneous Neumann boundary conditions
\begin{equation}\label{1.2}
\frac{\partial u}{\partial \eta}=\frac{\partial v}{\partial \eta}
=\frac{\partial w}{\partial \eta}=0\quad  \text{on }
 \mathbb{R}^{+}\times\partial \Omega  ,
\end{equation}
and initial data
\begin{equation}\label{1.3}
u(0,x)=\varphi _1(x),\quad v(0,x)=\varphi _2(x), \quad
 w(0,x)=\varphi_{3}(x),\quad \text{in}\;
\Omega.
\end{equation}

Here $\Omega$ is an open bounded domain in $\mathbb{R}^{N}$ with smooth
boundary $\partial \Omega$ and outer normal $\eta(x)$.
The constants $c,p_{i}, q_{i}, r_{i}, a_{i}$ and $b_{i}$, $i=1,2,3$ are
real numbers such that
\begin{equation*}
c,p_{i}, q_{i}, r_{i}\geq 0, \quad\text{and} \quad  a_{i}, b_{i}>0,
 \end{equation*}
 and
\begin{equation}\label{con}
0<p_1-1<\max\big\{ p_2\min\big( \frac{q_1}{q_2+1},
\frac{r_1}{r_2}, 1\big),
p_{3}\min\big(\frac{r_1}{r_{3}+1}, \frac{q_1}{q_{3}},
1\big)\big\}.
\end{equation}
The initial data are assumed to be positive and continuous functions
 on $\bar{\Omega}$. For $i=1,2,3$, we assume that $\sigma_{i}$ are
positive functions in $C(\bar{\Omega})$, and $\rho_{i}$ are positive bounded
functions in $C^{1}(\bar{\Omega}\times \mathbb{R}^{3}_{+})$.

In 1972, following the ingenious idea of Turing \cite{16}, Gierer
and Meinhardt \cite{3} proposed a mathematical model for pattern
formations of spatial tissue structure of hydra in morphogenesis, a
biological phenomenon discovered by Trembley in 1744 \cite{15}. It
can be expressed in the following system
\begin{equation}\label{*}
\begin{gathered}
\frac{\partial u}{\partial t}=a_1\Delta u -\mu_1u+\frac{u^{p}}{v^{q}}+\sigma,
\quad  \text{in } \mathbb{R}^{+}\times \Omega,\\
\frac{\partial v}{\partial t}=a_2\Delta v-\mu_2v+\frac{u^{r}}{v^{s}},
\quad  \text{in } \mathbb{R}^{+}\times \Omega,
\end{gathered}
\end{equation}
on a bounded  $\Omega\subset \mathbb{R}^{N}$, with the homogeneous
Neumann boundary conditions and positive initial data:
 $a_1,  a_2, \mu _1,\mu_2$ and $\sigma$ are positive constants, and $p,  q,  r,  s$
are non negative constants satisfying the relation
\begin{equation*}
\frac{p-1}{r}< \frac{q}{s+1}.
\end{equation*}


The  existence of global solutions to the system \eqref{*} is proved by
Rothe \cite{13} with special cases $N=3, p=2, q=1, r=2$ and $s=0$.
The Rothe's method can not be applied (at least directly) to general $p, q,  r, s$.
Wu and Li \cite{7} obtained the same results for the problem \eqref{*}
so long as $u,  v^{-1}$ and $\sigma$ are suitably small.
Li, Chen and Qin \cite{10} showed that the solutions of this problem are bounded
all the time for each pair of initial values in $L^{\infty}(\Omega)$ if
\begin{equation}\label{C1}
\frac{p-1}{r}<\min\big\{1,\frac{q}{s+1}\big\}.
\end{equation}
Masuda and Takahashi \cite{8} considered the generalized
Gierer-Meinhardt system
\begin{equation}\label{masuda}
\frac{\partial u_{i}}{\partial t}=a_{i}\Delta u_{i} -\mu_{i}u_{i}+g_{i}(x,u_1,u_2),
\quad \text{in } \mathbb{R}^{+}\times \Omega\;  ( i=1,2) ,
\end{equation}
where $a_{i},  \mu_{i}$,  $i=1,2$ are positive constants, and
\begin{gather*}
g_1(x,u_1,u_2)=\rho_1(x,u_1,u_2)\frac{u_1^{p}}{u_2^{q}}+\sigma_1(x),\\
g_2(x,u_1,u_2)=\rho_2(x,u_1,u_2)\frac{u_1^{r}}{u_2^{s}}+\sigma_2(x),
\end{gather*}
with $\sigma_1(\cdot)$ (resp. $\sigma_2(\cdot)$) is a positive (resp. non-negative)
$ C^{1}$ function on $\bar{\Omega}$, and $\rho_1$ (resp. $\rho_2$) is a non
negative (resp. positive) bounded and $C^{1}$ function on
$\bar{\Omega}\times \mathbb{R}^2_{+}$.
They extended the result of global existence of solutions for
\eqref{masuda} of Li, Chen and Qin \cite{10} to
\begin{equation}\label{n}
\frac{p-1}{r}<\frac{2}{N+2},
\end{equation}
and
\begin{equation}\label{cond init}
\begin{gathered}
\varphi_1, \varphi_2\in W^{2,l}(\Omega),\quad  l>\max \{ N,2\},\\
\frac{\partial \varphi_1}{\partial\eta}=\frac{\partial \varphi_2}{\partial\eta}=0
\quad \text{on } \partial\Omega\quad
\text{and}\quad\varphi_1\geq 0, \varphi_2>0 \quad \text{in }\bar{\Omega}.
\end{gathered}
\end{equation}

Jiang \cite{6} obtained the same results as Masuda and Takahashi
\cite{8} by another method such that \eqref{C1} and \eqref{cond init} are satisfied.

Abdelmalek, et al \cite{ll}  considered the following
Gierer-Meinhardt system of three equations
\begin{equation}\label{1.1.}
\begin{gathered}
\frac{\partial u}{\partial t}-a_1\Delta u=-b_1u+\frac{u^{p_1}}{v^{q_1}(w^{r_1}+c)}
+\sigma,\quad \text{in }\mathbb{R}^{+}\times\Omega,\\
\frac{\partial v}{\partial t}-a_2\Delta v=-b_2v+\frac{u^{p_2}}{v^{q_2}w^{r_2}},
\quad  \text{in } \mathbb{R}^{+}\times\Omega,\\
\frac{\partial w}{\partial t}-a_{3}\Delta w=-b_{3}w
 +\frac{u^{p_{3}}}{v^{q_{3}}w^{r_{3}}},\quad \text{in } \mathbb{R}^{+}\times\Omega,
\end{gathered}
\end{equation}
with homogeneous Neumann boundary conditions
\begin{equation}\label{1.2.}
\frac{\partial u}{\partial \eta}=\frac{\partial v}{\partial \eta}
=\frac{\partial w}{\partial \eta}=0\quad  \text{on }
 \mathbb{R}^{+}\times\partial \Omega  ,
\end{equation}
and the initial data
\begin{equation}\label{1.3.}
\begin{gathered}
 u(0,x)=\varphi _1(x)>0,\\
v(0,x)=\varphi _2(x)>0,\\
 w(0,x)=\varphi_{3}(x) >0
\end{gathered}
\end{equation}
in $\Omega$, and $\varphi_{i}\in C(\bar{\Omega})$ for all $i=1,2,3$.
Under the condition \eqref{con} and by using a suitable Lyapunov functional,
they studied the global existence of solutions for the system
\eqref{1.1.}--\eqref{1.3.}. Their method gave only the existence of global
 solutions, and they did not obtain results about
the uniform boundedness of solutions on $ (0,+\infty) $.

For the asymptotic behavior of the solutions, Wu and Li \cite{7}
considered the system \begin{equation}
\begin{gathered}
 \frac{\partial u_1}{\partial t}=a_1\Delta u_1-u_1
+\frac{u_1^{p}}{u_2^{q}}+\sigma _1( x) ,\quad
\text{in }\mathbb{R}^{+}\times \Omega , \\
\tau \frac{\partial u_2}{\partial t}=a_2\Delta u_2-u_2+\frac{
u_1^{r}}{u_2^{s}}+\sigma _2( x) ,\quad
\text{in }\mathbb{R}^{+}\times \Omega ,
\end{gathered}  \label{Wu}
\end{equation}
 with the constant of relaxation time $\tau>0$, and they proved that if
$\sigma_1\equiv\sigma_2\equiv 0$ and $\tau >\frac{q}{p-1}$, then
 $( u(t,x),v(t,x)) \to  (0,0)$ uniformly on
$\bar{\Omega}$ as $t\to +\infty$.

Under suitable conditions on $\tau$ and on the initial data,
Suzuki and Takagi \cite{14.,14..} also studied the behavior of the solutions
for \eqref{Wu} with the constant of relaxation time $\tau$.


We first treat the uniform boundedness of the solutions for Gierer-Meinhardt
system of three equations by proving that the Lyapunov function argument
proposed in \cite{ll} can be adapted to our situation.
Interestingly, we show that the same Lyapunov function satisfies
a differential inequality from which the uniform boundedness of the
solutions is deduced for any positive time.
Then under reasonable conditions on the coefficients $b_1,
b_2$ and $b_{3}$, and by using the uniform boundedness of the
solutions and the Lyapunov function which is non-increasing
function,  we deal with the long-time behavior of solutions as the
time goes to $+\infty$. In particular we are concerned with
$\sigma_1\equiv 0$, $\sigma_2$ and $ \sigma_{3}$ are
non-negative constants to assure that
\[
{\lim_{t\to +\infty}}\| u(t,.)\| _{\infty}
={\lim_{t\to +\infty}}\|  v(t,.)-\frac{\sigma_2}{b_2}\| _{\infty}
={\lim_{t\to +\infty}}\|  w(t,.)-\frac{\sigma_{3}}{b_{3}}\| _{\infty}=0\,.
\]

\section{Notation and preliminary results}

\subsection{Existence of local solutions}
For $i=1, 2, 3$ we set
\begin{gather*}
\b{$\varphi_{i}$}={\min_{x\in \bar{\Omega}}}\varphi_{i} (x),\quad
 \bar{\varphi_{i}}={\max_{x\in \bar{\Omega}}}\varphi_{i}(x),\\
\b{$\rho_{i}$}={\min_{x\in \bar{\Omega},\xi\in \mathbb{R}^{3}_{+}}}
\rho_{i}(x,\xi),\quad
\bar{\rho_{i}}={\max_{x\in \bar{\Omega},\xi\in \mathbb{R}^{3}_{+}}}
\rho_{i}(x,\xi) ,\\
\b{$\sigma_{i}$}= {\min_{x\in \bar{\Omega}}} \sigma_{i}(x),\quad
 \bar{\sigma_{i}}={\max_{x\in\bar{\Omega}}} \sigma_{i}(x).
\end{gather*}
The basic existence theory for abstract semi linear differential
equations directly leads to a local existence result to system
\eqref{1.1}--\eqref{1.3} (see, Henry \cite{5}). All solutions  are
classical on $( 0,T)\times \Omega$, $T<T_{\rm max}$, where
$T_{\rm max}(\lVert u_{0}\rVert_{\infty},\lVert w_{0}\rVert_{\infty})$
denotes the eventual blowing-up time in $L^{\infty}(\Omega)$.

\subsection{Positivity of solutions}

\begin{lemma}\label{lem1}
If $( u,v,w) $ is a solution of the problem \eqref{1.1}--\eqref{1.3},
then for all $(t,x)\in ( 0,T_{
m max}) \times \Omega $, we have
\begin{itemize}
\item[(1)]
\begin{gather*} %\label{4.5}
u(t,x)\geq e^{-b_1t}\underline{\varphi_1} >0,\\
v(t,x)\geq e^{-b_2t}\underline{\varphi_2} >0,\\
w(t,x)\geq e^{-b_{3}t}\underline{\varphi_3} >0.
\end{gather*}

\item[(2)]
\begin{gather*}
u(t,x)\geq \min\big( \underline{\sigma_1}\,/b_1,\underline{\varphi_1}\big) =m_1,\\
v(t,x)\geq \min\big( \underline{\sigma_2}\,/b_2 ,\underline{\varphi_2}\big)=m_2,\\
w(t,x)\geq \min\big( \underline{\sigma_3}\,/b_{3} ,\underline{\varphi_3}\big)=m_{3}.
\end{gather*}
\end{itemize}
\end{lemma}

The proof of the above lemma follows immediate from the maximum principle.

\section{Boundedness of solutions}
For proving the  existence of global solutions for  \eqref{1.1}--\eqref{1.3},
it suffices to prove that the solutions remains bounded in $(0,T)\times\bar{\Omega}$.
One of the main results of this paper reads as follows.

\begin{theorem}\label{th1}
Assume that \eqref{con} holds.
Let $( u, v, w) $ be a solution to \eqref{1.1}--\eqref{1.3}, and let
\begin{equation}\label{1.8}
L(t)={\int_{\Omega}}\frac{u^{\alpha}(t,x)}{v^{\beta}(t,x)w^{\gamma}(t,x)}dx,\quad
\text{for all } t\in (0,T),
\end{equation}
where $\alpha,  \beta$ and $\gamma$ are positive constants satisfying the
following conditions:
\begin{equation}\label{c1}
\quad\alpha>2\max\Big(1, \frac{3b_2+ b_{3}}{b_1}\Big),
\quad
\frac{1}{\beta}>\frac{(a_1+a_2)^2}{2a_1a_2},
\end{equation}
and
\begin{equation}\label{c2}
\Big(\frac{1}{2\beta}-\frac{(a_1+a_2)^2}{4a_1a_2}\Big)
\Big(\frac{1}{2\gamma}-\frac{(a_1+a_{3})^2}{4a_1a_{3}}\Big)
>\Big(\frac{(\alpha-1)(a_2+a_{3})}{2\alpha
\sqrt{a_2a_{3}}}-\frac{(a_1+a_2)(a_1+a_{3})}{4\sqrt{a_1^2a_2a_{3}}}\Big)^2.
\end{equation}
Then there exists a positive constant $C$ such that for all $t\in (0,T)$,
\begin{equation}\label{ly}
\frac{d}{dt} L(t)\leq -( \alpha b_1-3b_2\beta-\gamma b_{3}) L(t)+C.
\end{equation}
\end{theorem}

\begin{corollary}\label{coro}
Under the assumptions of Theorem \ref{th1}, all solutions of
\eqref{1.1}--\eqref{1.3} with positive initial data in $C(\bar{\Omega})$
are global and uniformly bounded on $(0,+\infty)\times \bar{\Omega}$.
\end{corollary}

Before proving the above theorem we first need the following technical lemma.

\begin{lemma}\label{l2}
Suppose that $x>0$,  $y>0$ and $z>0$, then for each group of indices
$r,  p,  q, \delta,  \theta,  \lambda$ and $\xi$ satisfies
$\lambda<p<\delta$ (not necessarily positive), and any constant
$\Lambda>0$, we have
\begin{equation*}
\frac{x^{p}}{y^{q}z^{r}}
\leq \Lambda\frac{x^{\delta}}{y^{\theta}z^{\xi}}
+\Lambda^{-\frac{p-\lambda}{\delta-p}}\frac{x^{\lambda}}{y^{\eta_1}z^{\eta_2}},
\end{equation*}
where
\begin{gather*}
\eta_1 = [q(\delta-\lambda)-\theta(p-\lambda)](\delta-p)^{-1},\\
\eta_2 = [r(\delta-\lambda)-\xi(p-\lambda)](\delta-p)^{-1}.
\end{gather*}
\end{lemma}

\begin{proof}
We can write
\[
\frac{x^{p}}{y^{q}z^{r}}
=\Big( x^{\frac{\delta(p-\lambda)}{\delta-\lambda}}
y^{-\frac{\theta(p-\lambda)}{\delta-\lambda}}
z^{-\frac{\xi(p-\lambda)}{\delta-\lambda}}\Big)
\Big( x^{\frac{\lambda(\delta-p)}{\delta-\lambda}}
y^{\frac{\theta(p-\lambda)}{\delta-\lambda}-q}z^{\frac{\xi(p-\lambda)}
{\delta-\lambda}-r}\Big).
\]
By using Young's inequality we obtain
\begin{equation*}
\frac{x^{p}}{y^{q}z^{r}}\leq \varepsilon
\frac{x^{\delta}}{y^{\theta}z^{\xi}}
+\varepsilon^{-\frac{p-\lambda}{\delta-p}}
\frac{x^{\lambda}}{y^{\eta_1}z^{\eta_2}},
\end{equation*}
where
\begin{gather*}
\eta_1=[q(\delta-\lambda)-\theta(p-\lambda)](\delta-p)^{-1},\\
\eta_2=[r(\delta-\lambda)-\xi(p-\lambda)](\delta-p)^{-1}.
\end{gather*}
Then Lemma \ref{l2} is  proved.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th1}]
Let $(u,v,w)$ be the solution of system \eqref{1.1}--\eqref{1.3} in $(0,T)$.
 Differentiating $L(t)$ respect to $t$, we obtain
$L'(t)=I+J$,
where
\begin{gather*}
I=a_1\alpha{\int _{\Omega}}
\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}}\Delta u \,dx
-a_2\beta\int _{\Omega}
\frac{u^{\alpha}}{v^{\beta+1}w^{\gamma}}\Delta v
\,dx-a_{3}\gamma\int _{\Omega}
\frac{u^{\alpha}}{v^{\beta}w^{\gamma+1}}\Delta w \,dx,
\\
\begin{aligned}
J&=( -\alpha b_1+\beta b_2 +\gamma b_{3}) L(t)
 +\alpha {\int _{\Omega}}\rho_1(x,u,v,w)
 \frac{u^{\alpha -1+p_1}}{v^{\beta +q_1}w^{\gamma+r_1}} dx \\
&\quad-\beta {\int _{\Omega}}\rho_2(x,u,v,w)
 \frac{u^{\alpha+p_2}}{v^{\beta+1+q_2}w^{\gamma+r_2}} dx
 -\gamma{\int_{\Omega}}\rho_{3}(x,u,v,w)
 \frac{u^{\alpha+p_{3}}}{v^{\beta+q_{3}}w^{\gamma+1+r_{3}}}  dx\\
&\quad +\alpha {\int _{\Omega}}\sigma_1(x)
 \frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}}dx
 -\beta{\int _{\Omega}} \sigma_2(x)\frac{u^{\alpha}}{v^{\beta+1}w^{\gamma}}  dx
 -\gamma{\int_{\Omega}} \sigma_{3}(x)\frac{u^{\alpha}}{v^{\beta}w^{\gamma+1}}dx.
\end{aligned}
\end{gather*}
Using Green's formula, for all $t\in (0,T)$, we obtain (see \cite{ll})
\begin{equation}\label{3.5.}
I\leq 0.
\end{equation}

Now let us estimate the term $J$. For all $t\in(0,T)$ we have
\begin{equation}\label{3.5}
\begin{aligned}
 J&\leq\big( -\alpha b_1+\beta b_2 +\gamma b_{3}\big) L(t)
+\alpha\bar{\rho_1} {\int _{\Omega}}
\frac{u^{\alpha -1+p_1}}{v^{\beta +q_1}w^{\gamma+r_1}} dx
-\beta \underline{\rho_2} {\int _{\Omega}}\frac{u^{\alpha+p_2}}{v^{\beta+1+q_2}
 w^{\gamma+r_2}}dx\\
&\quad -\underline{\rho_3}\gamma{\int_{\Omega}}
\frac{u^{\alpha+p_{3}}}{v^{\beta+q_{3}}w^{\gamma+1+r_{3}}} dx
+\alpha\bar{\sigma_1} {\int _{\Omega}}\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}}dx.
\end{aligned}
\end{equation}
Applying Lemma \ref{l2} with $p=\alpha-1$,  $q=\theta=\beta$,  $r=\gamma$,
$\delta=\alpha$, $\xi=\gamma$ and $\lambda=0$, one gets
\begin{equation}\label{3.6}
\alpha\bar{\sigma_1}{\int_{\Omega}}
\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}}dx
\leq \beta b_2{\int_{\Omega}}\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}dx
+C_1{\int_{\Omega}}\frac{1}{v^{\beta}w^{\gamma}}dx,
\end{equation}
where $C_1 =\alpha \bar{\sigma_1}( \frac{\beta b_2}{\alpha\bar{\sigma_1}}) ^{1-\alpha}$.\\
Now, we choose $\epsilon_1\in (0,\alpha)$ such that
\begin{gather*}
\beta+\alpha\frac{q_1p_2-(p_1-1)(1+q_2)}{\epsilon_1(p_2+1-p_1)}
+\alpha\frac{q_1-1-q_2}{p_2+1-p_1}\geq 0,\\
\gamma+\alpha \frac{r_1p_2-r_2(p_1-1)}{\epsilon_1(p_2-p_1+1)}
+\alpha\frac{r_1-r_2}{p_2-p_1+1} \geq 0.
\end{gather*}
Again, applying Lemma \ref{l2} for $p=\alpha-1+p_1$,  $q=\beta+q_1$,
$r=\gamma+r_1$,  $\delta=\alpha+p_2$,  $\theta=\beta+1+q_2$,  $\xi=\gamma+r_2$ and
$\lambda=\alpha-\epsilon_1$, we obtain
\begin{equation}
\alpha\bar{\rho_1}{\int_{\Omega}}
\frac{u^{\alpha-1+p_1}}{v^{q_1+\beta}w^{r_1+\gamma}}dx
\leq \beta\underline{\rho_2}{\int_{\Omega}} \frac{u^{p_2+\alpha}}{v^{q_2+\beta+1}
w^{r_2+\gamma}}dx
+C_2{\int_{\Omega}}\frac{u^{\alpha-\epsilon_1}}{v^{\eta_1}w^{\eta_2}}dx,
\end{equation}
 where
\begin{gather*}
\eta_1=\beta+[ q_1p_2-(q_2+1)(p_1-1)
 +\epsilon_1(q_1-q_2-1)] (p_2-p_1+1)^{-1},\\
\eta_2=\gamma+[ r_1p_2-r_2(p_1-1)
 +\epsilon_1(r_1-r_2)] (p_2-p_1+1)^{-1},
\end{gather*}
and $C_2=\alpha\bar{\rho_1}( \frac{\beta\underline{\rho_2}}{\alpha\bar{\rho_1}}
)^{-\frac{p_1-1+\epsilon_1}{p_2-p_1+1}}$.

In an analoguous way, we have
\begin{equation}
C_2{\int_{\Omega}}\frac{u^{\alpha-\epsilon_1}}{v^{\eta_1\eta_2}}dx
\leq b_2\beta{\int_{\Omega}} \frac{u^{\alpha}}{v^{\beta}w^{\gamma}}dx
+C_{3}{\int_{\Omega}}\frac{1}{v^{\eta_{3}\eta_{4}}}dx,
\end{equation}
where
\begin{gather*}
\eta_{3}=\beta+\alpha[\epsilon_1^{-1} (q_1p_2-(q_2+1)(p_1-1))
 +q_1-q_2-1]( p_2-p_1+1)^{-1}\geq 0,\\
\eta_{4}=\gamma+\alpha[ \epsilon_1^{-1}(r_1p_2-r_2(p_1-1))
 +r_1-r_2] (p_2-p_1+1)^{-1}\geq0,
\end{gather*}
and $C_{3}=C_2( \frac{b_2\beta}{C_2})
^{-\frac{\alpha-\epsilon_1}{\epsilon_1}}$.

Or, we choose $\epsilon_2\in(0,\alpha)$ such that
\begin{gather*}
\beta+\alpha\frac{q_1p_{3}-q_{3}(p_1-1)}{\epsilon_2(p_{3}-p_1+1)}
 +\alpha\frac{q_1-q_{3}}{p_{3}-p_1+1} \geq  0,\\
\gamma+\alpha\frac{r_1p_{3}-(r_{3}+1)(p_1-1)}{\epsilon_2(p_{3}-p_1+1)}
+\alpha\frac{r_1-r_2-1}{p_{3}-p_1+1} \geq 0.
\end{gather*}

Now, applying Lemma \ref{l2} with $p=p_1+\alpha-1$,  $q=q_1+\beta$,
$r=r_1+\gamma$,  $\delta=p_{3}+\alpha$, $\theta=q_{3}+\beta$,
$\xi=r_{3}+\gamma+1$ and $\lambda=\alpha-\epsilon_2$, we find that
\begin{equation}
\alpha\bar{\rho_1}{\int_{\Omega}}\frac{u^{\alpha-1+p_1}}{v^{\beta+q_1}
w^{\gamma+r_1}}dx\leq\gamma\underline{\rho_3}
{\int_{\Omega}}\frac{u^{\alpha+p_{3}}}{v^{\beta+q_{3}}w^{\gamma+1+r_{3}}}dx
+C_{4}{\int_{\Omega}} \frac{u^{\alpha-\epsilon_2}}{v^{\eta_{5}}w^{\eta_{6}}}dx,
\end{equation}
where
\begin{gather*}
\eta_{5}=\beta+[q_1p_{3}-q_{3}(p_1-1)+\epsilon_2(q_1-q_{3})](p_{3}-p_1+1)^{-1},\\
\eta_{6}=\gamma+[r_1p_{3}-(r_{3}+1)(p_1-1)
+\epsilon_2(r_1-r_{3}-1)](p_{3}-p_1+1)^{-1},
\end{gather*}
and $C_{4}=\alpha\bar{\rho_1}
( \frac{\gamma\underline{\rho_3}}{\alpha\bar{\rho_1}})
^{-\frac{p_1-1+\epsilon_2}{p_{3}-p_1+1}}$.
 In the same way, we obtain
 \begin{equation}\label{3.10}
C_{4}{\int_{\Omega}}\frac{u^{\alpha-\epsilon_2}}{v^{\eta_{5}}w^{\eta_{6}}}dx
\leq b_2\beta{\int_{\Omega}}\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}dx
+ C_{5}{\int_{\Omega}}\frac{1}{v^{\eta_{7}}w^{\eta_{8}}}dx,
 \end{equation}
where
\begin{gather*}
\eta_{7}= \beta+\alpha[ \epsilon_2^{-1} (q_1p_{3}-q_{3}(p_1-1))+q_1-q_{3}](p_{3}-p_1+1)^{-1}\geq 0,\\
\eta_{8}=\gamma+\alpha[ \epsilon_2^{-1}(r_1p_{3}-(r_{3}+1)(p_1-1))+r_1-r_{3}-1](p_{3}-p_1+1)^{-1}\geq0,
\end{gather*}
and $C_{5}=C_{4}( \frac{b_2\beta}{C_{4}})
^{-\frac{\alpha-\epsilon_2}{\epsilon_2}}$.


From \eqref{3.5}--\eqref{3.10} there exists a positive constant $C$ such that
\begin{equation*}
L'(t)\leq -(b_1\alpha-3\beta b_2-\gamma b_{3})L(t)+C,\quad \forall t\in (0,T).
\end{equation*}
Then the proof is complete.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro}]
Since
\begin{equation*}
L(t)\leq L(0)+\frac{C}{\alpha b_1-3b_2\beta-\gamma b_{3}}\quad
\text{for all } t\in (0,T),
\end{equation*}
then there exist non-negative constants $C_{6}$, $C_{7}$ and $C_{8}$
independent of $t$ such that
\begin{gather*}
\| f(u,v,w)-b_1u\|_{N} \leq C_{6},\\
\| g(u,v;w)-b_2v\|_{N} \leq  C_{7},\\
\| h(u,v,w)-b_{3}w\|_{N} \leq  C_{8}.
\end{gather*}

Since $(\varphi_1,\varphi_2,\varphi_{3})\in (C(\bar{\Omega}))^{3}$,
we conclude from the $L^{p}$-$L^{q}$-estimate (see Henry \cite{5},
Haraux and Kirane \cite{4}) that
\begin{equation*}
u\in L^{\infty}( (0,T),L^{\infty}(\Omega)) ,\quad
v\in L^{\infty}( (0,T),L^{\infty}(\Omega)),\quad
w\in L^{\infty}( (0,T),L^{\infty}(\Omega)).\\
\end{equation*}
Finally, we deduce that the solutions of the system \eqref{1.1}--\eqref{1.3}
are global and uniformly bounded on $(0,+\infty)\times\bar{\Omega}$.
\end{proof}


\begin{remark} \rm
It is clear that  the results of this section are valid when
 $\sigma_1\equiv\sigma_2\equiv\sigma_{3}\equiv 0$.
\end{remark}

\section{Asymptotic behavior of the solutions}

In this section, we  study the asymptotic behavior of the
solutions for the  system
\begin{equation}\label{4.1}
\begin{gathered}
\frac{\partial u}{\partial t}-a_1\Delta u=-b_1u+f(u,v,w),
 \quad \text{in } \mathbb{R}^{+}\times\Omega,\\
\frac{\partial v}{\partial t}-a_2\Delta v=-b_2v+g(u,v,w),
 \quad \text{in } \mathbb{R}^{+}\times\Omega,\\
\frac{\partial w}{\partial t}-a_{3}\Delta w=-b_{3}w+h(u,v,w),
 \quad \text{in } \mathbb{R}^{+}\times\Omega,
\end{gathered}
\end{equation}
where
\begin{gather*}
f(u,v,w)=\rho_1(x,u,v,w)\frac{u^{p_1}}{v^{q_1}(w^{r_1}+c)}+\sigma_1,\\
g(u,v,w)=\rho_2(x,u,v,w)\frac{u^{p_2}}{v^{q_2}w^{r_2}}+\sigma_2,\\
h(u,v,w)=\rho_{3}(x,u,v,w)\frac{u^{p_{3}}}{v^{q_{3}}w^{r_{3}}}+\sigma_{3},
\end{gather*}
with homogeneous Neumann boundary conditions
\begin{equation}
\frac{\partial u}{\partial \eta}=\frac{\partial v}{\partial \eta}
=\frac{\partial w}{\partial \eta}=0\quad  \text{on }
 \mathbb{R}^{+}\times\partial \Omega  ,
\end{equation}
and initial data
\begin{equation}\label{4.4}
u(0,x)=\varphi _1(x),\quad v(0,x)=\varphi _2(x),\quad
w(0,x)=\varphi_{3}(x)\quad \text{in } \Omega.
\end{equation}\\
Here $\sigma_1$,  $\sigma_2$ and $\sigma_{3}$ are non negative constants.

Before stating the results, let us expose some simple facts concluded from
the result of the previous section.
From Theorem \ref{th1}, and by using classical method of a semi
group and a power fractional (see \cite{5}) we can find the positive
constants $M_1,  M_2$ and $M_{3}$ explicitly (see \cite{mmy})
such that
\[
\| u(t,.)\|_{\infty} M_1,\quad
\| v(t,.)\| _\infty \leq M_2,\quad
\| w(t,.)\| _\infty \leq M_{3}.
\]

Let us consider the same function as in Theorem \ref{th1},
\begin{equation*}
L(t)={\int _{\Omega}}\frac{u^{\alpha}(t,x)}{v^{\beta}(t,x)w^{\gamma}(t,x)}dx,\quad
 \forall t\in (0,+\infty),
\end{equation*}
where $\alpha,  \beta$ and $\gamma$ are positive constants satisfying
the following conditions
\[% \label{c1}
\quad\alpha>2\max(1, \frac{3b_2+ b_{3}}{b_1}),
\quad
\frac{1}{\beta}>\frac{(a_1+a_2)^2}{2a_1a_2},
\]
and
\[ %\label{c2}
\Big(\frac{1}{2\beta}-\frac{(a_1+a_2)^2}{4a_1a_2}\Big)
\Big(\frac{1}{2\gamma}-\frac{(a_1+a_{3})^2}{4a_1a_{3}}\Big)
>\Big(\frac{(\alpha-1)(a_2+a_{3})}{2\alpha
\sqrt{a_2a_{3}}}-\frac{(a_1+a_2)(a_1+a_{3})}{4\sqrt{a_1^2a_2a_{3}}}\Big)^2.
\]
The main result in this section reads as follows.

\begin{theorem}\label{th2}
Assume \eqref{con} holds. Let  $(u,v,w)$ be the solution of
\eqref{4.1}--\eqref{4.4} in $(0,+\infty) $. Suppose that $\sigma_1=0$, and
\begin{equation}\label{conb}
b_1>\frac{\beta b_2+\gamma b_{3}+K}{2},
\end{equation}
where
\[
K=\frac{\alpha \bar{\rho_1}(\frac{\beta\underline{\rho_2}}
{\alpha\bar{\rho_1}})^{-\frac{p_1-1}{p_2-p_1+1}}}
{m_2^{[q_1p_2-(q_2+1)(p_1-1)](p_2-p_1+1)^{-1}}m_{3}^{[r_1p_2-r_2(p_1-1)]
(p_2-p_1+1)^{-1}}},
\]
or
\begin{equation*}
K=\frac{\alpha \bar{\rho_1}
(\frac{\gamma\underline{\rho_3}}{\alpha\bar{\rho_1}})
^{-\frac{p_1-1}{p_{3}-p_1+1}}}{{m_2^{[q_1p_{3}-q_{3}(p_1-1)]
(p_{3}-p_1+1)^{-1}}m_{3}^{[r_1p_{3}-(r_{3}+1)(p_1-1)](p_{3}-p_1+1)^{-1}}}}.
\end{equation*}
Then for all $t\in(0,+\infty)$ we have
\[
L(t)\leq {\int_{\Omega}}\frac{\varphi_1^{\alpha}(x)}
{\varphi_2^{\beta}(x)\varphi_{3}^{\gamma}(x)}dx.
\]
\end{theorem}

\begin{corollary}\label{coro2}
Under the assumptions of Theorem \ref{th2}, for all positive initial data
in $C(\bar{\Omega})$ we have
\begin{gather*}
\| u(t,.)\| _{\infty}\to  0 \quad \text{as }t\to +\infty, \\
\|  v(t,.)-\frac{\sigma_2}{b_2}\| _{\infty}\to  0\quad\text{as }t\to +\infty,\\
\|  w(t,.)-\frac{\sigma_{3}}{b_{3}}\| _{\infty}\to  0\quad \text{as }t\to +\infty.
\end{gather*}
\end{corollary}


\begin{proof}[Proof of Theorem \ref{th2}]
From \eqref{3.5.} and \eqref{3.5}, we obtain for all $t\in (0,+\infty)$
\begin{equation}\label{4.8}
\begin{aligned}
 L'(t)&\leq -(\alpha b_1-\beta b_2-\gamma b_2)L(t)
 +\alpha\bar{\rho_1}{\int_{\Omega}}\frac{u^{\alpha-1+p_1}}{v^{\beta+q_1}
w^{\gamma+r_1}}dx\\
&\quad -\beta \underline{\rho_2} {\int_{\Omega}}
\frac{u^{\alpha+p_2}}{v^{\beta+1+q_2}w^{\gamma+r_2}}dx
-\gamma \underline{\rho_3}{\int_{\Omega}}\frac{u^{\alpha+p_{3}}}
{v^{\beta+q_{3}}w^{\gamma+1+r_{3}}}dx.
\end{aligned}
\end{equation}
Now, we apply Lemma \ref{l2} for $p=\alpha-1+p_1$,  $q=\beta+q_1$,
$ r=\gamma+r_1$,  $\delta=\alpha+p_2$,  $\theta=\beta+1+q_2$,
$\xi=\gamma+r_2$ and $\lambda=\alpha$ we obtain
\begin{equation}\label{4.9}
\alpha\bar{\rho_1}{\int_{\Omega}}\frac{u^{\alpha-1+p_1}}{v^{\beta+q_1}
w^{\gamma+r_1}}dx \leq \beta\underline{\rho_2}{\int_{\Omega}}
\frac{u^{\alpha+p_2}}{v^{\beta+1+q_2}w^{\gamma+r_2}}dx
+A_1{\int_{\Omega}} \frac{u^{\alpha}}{v^{\eta_{9}}w^{\eta_{10}}}dx,
\end{equation}
where
\begin{gather*}
\eta_{9} = \beta +[q_1p_2-(q_2+1)(p_1-1)](p_2-p_1+1)^{-1}>0,\\
\eta_{10}= \gamma+ [ r_1p_2-r_2(p_1-1)](p_2-p_1+1)^{-1}>0,
\end{gather*}
and  $A_1=\alpha\bar{\rho_1}
( \frac{\beta\underline{\rho_2}}{\alpha\bar{\rho_1}})
^{-\frac{p_1-1}{p_2-p_1+1}}$.


Or, applying Lemma \ref{l2} for $p=\alpha-1+p_1$, $q=\beta+q_1$,
$r=\gamma+r_1$,  $\delta=\alpha+p_{3}$,  $\theta=\beta+q_{3}$,
$\xi=\gamma+1+r_{3}$ and $\lambda=\alpha$, we obtain
\begin{equation}\label{4.10}
\alpha\bar{\rho_1}{\int_{\Omega}}\frac{u^{\alpha-1+p_1}}
{v^{\beta+q_1}w^{\gamma+r_1}}dx
\leq\gamma\underline{\rho_3}{\int_{\Omega}}
\frac{u^{\alpha+p_{3}}}{v^{\beta+q_{3}}w^{\gamma+1+r_{3}}}dx
+ A_2{\int_{\Omega}}\frac{u^{\alpha}}{v^{\eta_{11}}w^{\eta_{12}}}dx,
\end{equation}
where
\begin{gather*}
\eta_{11} = \beta+[q_1p_{3}-q_{3}(p_1-1)](p_{3}-p_1+1)^{-1}>0,\\
\eta_{12} = \gamma+[r_1p_{3}-(r_{3}+1)(p_1-1)](p_{3}-p_1+1)^{-1}>0,
\end{gather*}
and $A_2=\alpha\bar{\rho_1}
( \frac{\gamma\underline{\rho_3}}{\alpha\bar{\rho_1}})
^{-\frac{p_1-1}{p_{3}-p_1+1}}$.
By combining \eqref{4.8} with \eqref{4.9} and \eqref{4.10} we obtain
\begin{equation}\label{4.11}
L'(t)\leq -(\alpha b_1-\beta b_2-\gamma b_{3}-K)L(t),\quad
\forall t\in (0,+\infty),
\end{equation}
where
\begin{equation*}
K=\frac{\alpha \bar{\rho_1}
(\frac{\beta\underline{\rho_2}}{\alpha\bar{\rho_1}}
)^{-\frac{p_1-1}{p_2-p_1+1}}}{m_2^{[q_1p_2-(q_2+1)
(p_1-1)](p_2-p_1+1)^{-1}}m_{3}^{[r_1p_2-r_2(p_1-1)](p_2-p_1+1)^{-1}}},
\end{equation*}
or
\begin{equation*}
K=\frac{\alpha \bar{\rho_1}(\frac{\gamma\underline{\rho_3}}
{\alpha\bar{\rho_1}})^{-\frac{p_1-1}{p_{3}-p_1+1}}}
{{m_2^{[q_1p_{3}-q_{3}(p_1-1)](p_{3}-p_1+1)^{-1}}m_{3}^{[r_1p_{3}-(r_{3}+1)(p_1-1)]
(p_{3}-p_1+1)^{-1}}}}.
\end{equation*}
Using \eqref{conb} we deduce that the function $t\longmapsto L(t)$
is a non-increasing function.
This completes the proof of Theorem \ref{th2}.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro2}]
Setting for all $(t,x)\in(0,+\infty)\times \Omega$:
\begin{gather*}
h_1(t,x)= u(t,x),\\
h_2(t,x)= v(t,x)-\frac{\sigma_2}{b_2},\\
h_{3}(t,x)= w(t,x)-\frac{\sigma_{3}}{b_{3}}.
\end{gather*}
For $i=1,2,3$ we have
\begin{equation}\label{5.8}
\frac{d h_{i}}{dt}-a_{i}\Delta h_{i}=-b_{i}h_{i}+\rho_{i}(x,u,v,w)
\frac{u^{p_{i}}}{v^{q_{i}}w^{r_{i}}}.
\end{equation}
 Multiplying \eqref{5.8} by $h_{i}(t,x)$, $i=1, 2, 3$ and integrating over
$[0,t]\times\Omega$ we obtain
\begin{align*}
&\frac{1}{2}{\int_{\Omega}}h_{i}^2dx
 +a_{i}{\int_{0}^{t}}{\int_{\Omega}}\vert \nabla h_{i}\vert ^2\,dx\,ds
+b_{i}{\int_{0}^{t}}{\int_{\Omega}}h_{i}^2\,dx\,ds\\
&= \frac{1}{2}{\int_{\Omega}}h_{i}^2(0)dx
+{\int_{0}^{t}}{\int_{\Omega}}h_{i}\rho_{i}(x,u,v)
 \frac{u^{p_{i}}}{v^{q_{i}}w^{r_{i}}}\,dx\,ds.
\end{align*}
From \eqref{4.11}, for all $ t\in (0,+\infty)$, and for  $i=1,2, 3$ we obtain
\[
{\int_{0}^{t}}{\int_{\Omega}}h_{i}\rho_{i}(x,u,v)\frac{u^{p_{i}}}{v^{q_{i}}
w^{r_{i}}}\,dx\,ds
\leq \bar{\rho_{i}}M_{i}\frac{M_1^{p_{i}}M_2^{\beta}M_{3}
^{\gamma}}{m_2^{q_{i}}m_1^{\alpha}m_{3}^{r_{i}}}
{\int_{0}^{t}}{\int_{\Omega}} \frac{u^{\alpha}}{v^{\beta}w^{\gamma}}\,dx\,ds<+\infty.
\]
One obviously deduces that for $i=1,2,3$,
\begin{gather*}
h_{i}(t,.)\in L^2(\Omega),\quad
 {\int_{0}^{+\infty}}{\int_{\Omega}}\vert \nabla h_{i}\vert^2 dx ds<+\infty,\\
{\int_{0}^{+\infty}}{\int_{\Omega}}h_{i}^2\,dx\,ds<+\infty,
\end{gather*}
so that Barbalate's lemma \cite[Lemma 1.2.2]{3.} permits to conclude that
\begin{equation*}
{\lim_{t\to +\infty}}\| h_{i}(t,.)\| _2=0,\quad i=1,2,3.
\end{equation*}
On the other hand, since the orbits $\{  h_{i}(t,.)/t\geq 0,  i=1,2,3\}  $
are relatively compact in $C(\bar{\Omega})$ (see \cite{4}), it follows readily that
\begin{equation*}
{\lim_{t\to +\infty}}\| h_{i}(t,.)\| _{\infty}=0,\quad i=1,2,3.
\end{equation*}
Then proof of Corollary \ref{coro2} is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors want to thank Prof. M. Kirane and the
anonymous referee for their suggestions that improved the 
quality of this article.


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\end{document}