\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 156, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/156\hfil 
 Invariant regions and existence of global solutions]
{Invariant regions and existence of global solutions to reaction-diffusion
systems without conditions on the growth of nonlinearities}

\author[S. Bendoukha, S. Abdelmalek \hfil EJDE-2016/156\hfilneg]
{Samir Bendoukha, Salem Abdelmalek}

\address{Samir Bendoukha \newline
Electrical Engineering Department,
 College of Engineering at Yanbu,
Taibah University, Saudi Arabia}
\email{sbendoukha@taibahu.edu.sa}

\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{sallllm@gmail.com}

\thanks{Submitted May 2, 2016. Published June 21, 2016.}
\subjclass[2010]{35K45, 35K57}
\keywords{Reaction-diffusion system; global solution, Lyapunov functional}

\begin{abstract}
 This article concerns the existence of global solutions for a coupled
 2-component reaction diffusion system with a full matrix diffusion and
 exponential nonlinearities. We show that some results of global and bounded
 solutions are established via invariant regions and the Lyapunov functional. A
 numerical example is used to illustrate our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}\label{SecModel}

Reaction-diffusion systems have received considerable attention from
mathematicians and other scientists and engineers alike because of their ability
to model real-life phenomena in a wide variety of fields.
The study of these systems has allowed for a deeper understanding of the dynamics and characteristics of the phenomena. In this article, we study the
generic reaction-diffusion system with a full diffusion-matrix,
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}-a\Delta u-b\Delta v=f(u,v) \quad\text{in }
\mathbb{R}^{+}\times\Omega\\
\frac{\partial v}{\partial t}-c\Delta u-d\Delta v=g(u,v)\quad\text{in }
\mathbb{R}^{+}\times\Omega,
\end{gathered} \label{1.1}
\end{equation}
with the boundary conditions
\begin{equation}
\frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=0, \label{1.2}
\end{equation}
and initial data
\begin{equation}
u(0,x)=u_0(x),\quad v(0,x)=v_0(x),\quad \text{in }\Omega. \label{1.3}
\end{equation}
Here $\Omega$ is an open bounded domain of class $\mathbb{C}^{1}$ in
$\mathbb{R}^{N}$, with boundary $\partial\Omega$ and $\frac{\partial}{\partial\eta}$
denotes the outward normal derivative on $\partial\Omega$. We will assume that
the nonlinearities $f$ and $g$ are continuously differentiable functions on
$\mathbb{R}^{+}$. The constants $a$, $b$, $c$ and $d$ are positive and satisfying the condition
\begin{equation}
(b+c) ^2<4ad, \label{ParaCond}
\end{equation}
which reflects the parabolicity of the system and implies at the same time
that the diffusion matrix:
\begin{equation}
A=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}  \label{1.4}
\end{equation}
is positive definite. That means the eigenvalues $\lambda_1$ and
$\lambda_2$ $(\lambda_1<\lambda_2)  $ of $A^{T}$ are positive.

One of the earliest works on this subject is that of Kanel and Kirane
\cite{Kanel1998}, where they proved the  existence of global solutions given
$g(u;v)  =-f(u;v)  =-uv^{n}$ with $n$ being an odd
integer subject to
\[
\| b-c\| <C_{p},
\]
where $C_{p}$ contains a constant from Solonnikov's estimate. They further
improved their work in \cite{Kanel2000} and relaxed the conditions on the
diffusion matrix. They showed the existence of global solutions subject to
\begin{itemize}
\item[(H1)] $d<a+c$,

\item[(H2)]  $b<\varepsilon_0\equiv(\frac{ad(a+c-d)  }{ad+(a+c-d)  })  $ if $a\leq d<a+c$,

\item[(H3)] $b<\min\{  \frac{1}{2}(a+c)  ,\varepsilon_0\}$, and
\[
\| F(v)  \| \leq C_{F}(1+\| v\| ^{1+\varepsilon})  ,
\]
where $\varepsilon$ and $C_{F}$ are positive constants with
$\varepsilon<1$ being sufficiently small and $g(u;v)  =-f(u;v)=uF(v)  $.
\end{itemize}
Kouachi \cite{Kouachi2004} considered the case where the nonlinearities
$f(u,v)\geq0$ and $g(u,v)\geq0$ are continuously differentiable polynomially
bounded functions satisfying
\[
\mu_2g(\mu_2s,s)\leq f(\mu_2s,s),\quad \text{and}\quad
f(\mu_1s,s)\leq \mu_1g(\mu_1s,s),
\]
for all $s\geq0$, and
\[
f(u,v)+Cg(u,v)\leq C_1(u+\alpha v+1)  ,
\]
for positive $C,\alpha>-\mu_2$ and $C_1$ a positive constant and with
\begin{equation}
\mu_1=\frac{\min\{  a,d\}  -\lambda_1}{c}>0>\mu_2=\frac
{\min\{  a,d\}  -\lambda_2}{c}. \label{1.40}
\end{equation}
The author was able to determine the invariant regions of the system and
establish the existence of global solutions through an
appropriate Lyapunov functional.

Kouachi \cite{Kouachi2003} again considered the case that $f(u,v)=-\frac
{\rho}{\sigma}g(u,v)$ with $g(u,v)\geq0$, and a full diffusion-matrix with a
balance law. The study established the invariant regions of the system as well
as the existence of global solutions. The reaction term was assumed to be of
polynomial or sub-exponential growth. This work was later extended by Rebiai
and Benachour \cite{Rebiai2010}, where the authors relaxed the conditions on
the nonlinearities.

In our work, we will distinguish between the following two main cases for the
diffusion matrix $A$ and the corresponding invariant regions.
\smallskip

\noindent\textbf{Case 1: $c=0$.}
We assume that $a<d$ and that the initial data is in the region
\begin{equation}
\Sigma_1=\{  (u_0,v_0)  \in\mathbb{R}^2
\text{ such that }0\leq\frac{b}{d-a}v_0\leq u_0\}  .
\label{1.5}
\end{equation}
If $w=u-\frac{b}{d-a}v$ is uniformly bounded, we can suppose that for all
positive constants $M$, the nonlinearity $g$ is controlled for $v$ being
sufficiently large
\begin{equation}
0\leq g(u,v)  \leq H(v) ,\quad\text{for  }
0\leq u-\frac{b}{d-a}v\leq M, \label{1.6}
\end{equation}
where $H$ is a continuously differentiable function satisfying
\begin{gather}
\lim_{v\to\infty} \frac{H'(v)}{f(u,v)-\frac{b}{d-a}g(u,v)}=0,\quad
\text{for all }(u,v)  \in \Sigma_1, \label{1.7} \\
f(u,v)\leq\frac{b}{d-a}g(u,v)\text{ for all }(u,v)  \in\Sigma
_1, \label{1.7.5} \\
f(\frac{b}{d-a}v,v)\leq\frac{b}{d-a}g(\frac{b}{d-a}v,v), \label{350}
\end{gather}
with $g(u,0)\geq0$ for all $u\geq0$ and $v\geq0.$
\smallskip

\noindent\textbf{Case 2: $c\neq0$.}
The initial data are assumed to be in one of the following regions:
\begin{equation}
\begin{aligned}
&\Sigma_2=\{  (u_0,v_0)  \in\mathbb{R}^2\text{ such that }
\mu_2v_0\leq u_0\leq\mu_1v_0\} \\
&\Sigma_{3}=\{  (u_0,v_0)  \in\mathbb{R}^2\text{ such that }
 \mu_1v_0\leq u_0\leq\mu_2v_0\} \\
&\Sigma_{4}=\{  (u_0,v_0)  \in\mathbb{R}^2\text{ such that }
 \min(\mu_2v_0,\mu_1v_0)  \geq u_0\} \\
&\Sigma_{5}=\{  (u_0,v_0)  \in\mathbb{R}^2\text{ such that }
 u_0\geq\max(\mu_2v_0,\mu_1v_0)\}  ,
\end{aligned} \label{1.8}
\end{equation}
with $\mu_1$ and $\mu_2$ as defined in \eqref{1.40}.

In our work we will only deal with the first case ($\Sigma_2$).
Generalization to the remaining regions is trivial and can be looked up in the
appendix. We suppose that the reaction terms $f$ and $g$ satisfy:
\begin{gather}
\mu_1g(\mu_2s,s)\geq f(\mu_2s,s)\quad \text{for all }s\geq0, \label{1.12} \\
f(r,\frac{1}{\mu_1}r)\geq\mu_2g(r,\frac{1}{\mu_1}r),\quad \text{for all
}r\geq0,  \label{1.13} \\
f(r,s)\leq\mu_2g(r,s),\quad \text{for all }(r,s)  \in\Sigma_2\,.
\label{******}
\end{gather}
If $w=u-\mu_2v$ is uniformly bounded, we know that for all positive
constants $M$, the nonlinearity $g$ is controlled for $v$ being sufficiently
large, i.e. for $0\leq(\mu_1-\mu_2)  p-s\leq M$,
\begin{equation}
\mu_1g(\mu_1p-s,s)-f(\mu_1p-s,p)\leq H(s)  , \label{1.14}
\end{equation}
where $H$ is a continuously differentiable function satisfying
\begin{equation}
\lim_{s\to \infty} \frac{H'(s)  }
{f(\mu_1p-s,p)-\mu_2g(\mu_1p-s,p)}=0. \label{1.15}
\end{equation}
This class of systems motivates us to construct the type of functionals
considered in this paper with the aim of proving the existence of
global solutions.

\section{Invariant regions}

In this section, we are concerned with the invariant regions of the proposed
system. We will prove that if the pair $(f,g)  $ points into one
of the previously defined regions $\Sigma$ (either $\Sigma_1$, $\Sigma_2$,
$\Sigma_{3}$, $\Sigma_{4}$, or $\Sigma_{5}$) on $\partial\Sigma$, then
$\Sigma$ is an invariant region for problem \eqref{1.1}--\eqref{1.3}, i.e the
solution remains in $\Sigma$ for any initial data in $\Sigma$. Once the
invariant regions are constructed, one can apply the Lyapunov technique in
order to establish the global existence of unique solutions for the proposed
problem \eqref{1.1}--\eqref{1.3} as will be shown later on in Section
\ref{SecGlobal} (see for related examples the work of Kirane and Kouachi in
\cite{Kirane1993} and \cite{Kirane1996}).

\begin{proposition} \label{prop1}
Suppose that the functions $f$ and $g$  point into the region $\Sigma$ on
$\partial\Sigma$, then for any $(u_0,v_0)$ in $\Sigma,$ the solution
$(u(t,\cdot),v(t,\cdot))$ of problem \eqref{1.1}--\eqref{1.3} remains
in $\Sigma$ for any time $t\in[  0,T^{\ast}]  $.
\end{proposition}

\begin{proof}
We will approach the two cases discussed in Section \ref{SecModel} separately.
In the first case, $c=0$. Multiplying the second equation in \eqref{1.1} by
$\frac{b}{a-d}$ and adding the result to the first equation in \eqref{1.1}
yields the equivalent system
\begin{equation}
\begin{gathered}
\frac{\partial w}{\partial t}-a\Delta w=F(w,v)  =f(u,v)+\frac
{b}{a-d}g(u,v)\\
\frac{\partial v}{\partial t}-d\Delta v=g(u,v),
\end{gathered}  \label{2.1}
\end{equation}
where
\[
w=u+\frac{b}{a-d}v,
\]
and the initial data
\[
w(0,x)=w_0(x)\geq0,\quad  v(0,x)=v_0(x)\geq0\quad \text{in }\Omega,
\]
with the Neuman boundary conditions
\[
\frac{\partial w}{\partial\eta}=\frac{\partial v}{\partial\eta}=0.
\]
Using \eqref{350}, the first property is assured by the quasi-positivity of
the nonlinearities; that is:
\begin{equation}
F(0,v)  \geq0\text{ and }g(u,0)\geq0\quad \text{for all }w\geq0
\text{ and }v\geq0. \label{2.2}
\end{equation}

In the second case, $c\neq0$. It suffices to show that region $\Sigma_2$ is
invariant. The proof can be trivially extended to the other regions. We
construct a new system, which is equivalent to \eqref{1.1}. The first equation
is formed by multiplying the second equation in \eqref{1.1} by $\mu_1$ and
subtracting the second equation from it. The second is obtained by multiplying
the second equation in \eqref{1.1} by $-\mu_2$ and adding it to the first
one. Then, assuming without loss of generality that $a<d$ and with the fact
that $\lambda_1$ and $\lambda_2$ are the eingenvalues of $A$, we can
write
\begin{equation}
\begin{gathered}
\frac{\partial w}{\partial t}-\lambda_2\Delta w=k(w,z)  ,\\
\frac{\partial z}{\partial t}-\lambda_1\Delta z=h(w,z)  ,
\end{gathered}  \label{2.4}
\end{equation}
where
\begin{equation}
\begin{gathered}
w=-\mu_2v+u\geq0\\
z=\mu_1v-u\geq0,
\end{gathered} \label{2.5}
\end{equation}
the eigenvalues of $A$ are given by
\begin{equation}
\begin{gathered}
\lambda_1=\frac{1}{2}(a+d-\sqrt{(a-d)  ^2+4bc})\\
\lambda_2=\frac{1}{2}(a+d+\sqrt{(a-d)  ^2+4bc}),
\end{gathered}  \label{2.6}
\end{equation}
and
\begin{equation}
\begin{gathered}
k(w,z)=-\mu_2g(u,v)+f(u,v)\\
h(w,z)=\mu_1g(u,v)-f(u,v).
\end{gathered}  \label{351}
\end{equation}

Using \eqref{1.12} and \eqref{1.13}, the first property is assured by the
quasi-positivity of the nonlinearities; that is:
\begin{equation}
k(0,z)  \geq0,\ h(w,0)  \geq0\text{ for all }
w\geq0\text{ and }z\geq0. \label{2.7}
\end{equation}
Now using \eqref{1.14}, we can suppose that for all positive constants $M$,
the nonlinearity $g$ is controlled for $v$ being sufficiently large
\begin{equation}
h(w,z)  \leq H(z)  ,\quad\text{for }0\leq w\leq M, \label{2.8}
\end{equation}
where $H$ is a continuously differentiable function satisfying
\begin{equation}
\lim_{z\to \infty}\frac{H'(z)}{k(w,z)  }=0,\quad\text{for  }0\leq w\leq M.
\label{2.9}
\end{equation}
This concludes the proof of the proposition for regions $\Sigma_1$ and
$\Sigma_2$.
\end{proof}

\section{Existence of local solutions}\label{SecLocal}

In this section, we prove the existence of local solutions using
basic existence theory. First of all, let us define the usual norms in spaces
$L^p(\Omega)$, $L^{\infty}(\Omega)$, and $C(\overline{\Omega})$ as
\begin{gather*}
\| u\| _{p}^p=\frac{1}{\| \Omega\| }\int_{\Omega}\| u(x)\| ^pdx, \quad
\| u\| _{\infty}=\underset{x\in\Omega}{\operatorname{esssup}}\| u(x)\| , \\
\| u\| _{C(\overline{\Omega})}=\max_{x\in\overline {\Omega}} \| u(x)\| ,
\end{gather*}
respectively.

Now, for any initial data in 
$C(\overline{\Omega})\times C(\overline{\Omega })$ or in 
$L^p(\Omega)\times L^p(\Omega)$ with $p\in(1,+\infty)$, the  existence 
and uniqueness of local solutions to  \eqref{1.1}--\eqref{1.3} follow 
from the basic existence theory for abstract semilinear differential equations 
(see  Henry \cite{Henry1981}). 
Also note that the solutions are classical on 
$[0,T_{\rm max})  $ where $T_{\rm max}$ denotes the eventual blowing-up time in
$L^{\infty}(\Omega)$.

\section{Existence of global solutions}\label{SecGlobal}

In this section, we prove the existence of global solutions for the
diagonalized system \eqref{2.4}-\eqref{2.6}. Our main results are summarized
in the following theorem and corollary.

\begin{theorem}\label{TheoGlobal}
Let $(w(t,\cdot)  ,z(t,\cdot))  $ be any positive solution of the 
diagonal problem \eqref{2.4}-\eqref{2.6} on the interval $[  0,T]  $ f
or some $T<T^{\ast}$.
Then, assuming Neumann boundary conditions, the functional
\begin{equation}
t\to  L(t) =\int_{\Omega}(M-w)  ^{-\gamma
}H^p(z)  dx, \label{3.1}
\end{equation}
is uniformly bounded on $[0,T)  $ for any positive constants
$\gamma$, $M$ and $p$ satisfying
\begin{equation}
0<\gamma<\frac{4ab}{(\lambda_2-\lambda_1)  ^2},\quad
\| w(t,x) \| _{\infty}<M \quad \text{for all }0<t\leq T, \label{3.2}
\end{equation}
with
\begin{equation}
p>\frac{4(\gamma+1)  \lambda_1\lambda_2}{4\lambda_1
\lambda_2-\gamma(\lambda_2-\lambda_1)  ^2}. \label{3.3}
\end{equation}
\end{theorem}


\begin{proof} 
Differentiating the functional $L$ with respect to $t$ yields:
\begin{align*}
L'(t) &=\frac{d}{dt} \int_{\Omega}(M-w)  ^{-\gamma}H^p(z)  \,dx \\
&  =\int_{\Omega}\big[  H^p(z)  \frac{d}{dt}(M-w)  ^{-\gamma}
 +(M-w)  ^{-\gamma}\frac{d}{dt}H^p(z)  \big] \, dx\\
&  =\int_{\Omega}\Big\{  \gamma(M-w)  ^{-\gamma-1}H^p(
z)  [  \lambda_2\Delta w+k(w,z)  ]   \\
&\quad  +p(M-w)  ^{-\gamma}H^{p-1}H'[\lambda_1\Delta z+h(w,z)]
\Big\} \, dx\\
&  =I+J,
\end{align*}
where
\begin{gather*}
I=\int_{\Omega}\big\{  \lambda_2\gamma(M-w)  ^{-\gamma-1}
H^p(z)  \Delta w+\lambda_1p(M-w)  ^{-\gamma
}H^{p-1}H'\Delta z\big\} \, dx, \\
J=\int_{\Omega}(M-w)  ^{-\gamma-1}H^p(z)  \big\{
\gamma k(w,z)  +p(M-w)  \frac{H'}{H}h(w,z)  \big\} \, dx.
\end{gather*}
A simple application of Green's formula in \eqref{1.2} to $I$ yields
\[
I:=I_1+I_2.
\]
Simplifying the first part of $I$ leads to
\begin{align*}
I_1  
&  =-\int_{\Omega}\nabla [  \lambda_2\gamma(M-w)
^{-\gamma-1}H^p(z)  ]  \nabla w\,dx\\
&  =-\lambda_2\gamma\int_{\Omega}\Big\{  (\gamma+1)  (
M-w)  ^{-\gamma-2}H^p(z)  \| \nabla w\|^2 \\
&  \quad +p(M-w)^{-\gamma-1}H^{p-1}(z)  H'\nabla z\nabla w\Big\}\,  dx,
\end{align*}
and similarly, the second part can be simplified to produce
\begin{align*}
I_2  &  =-\int_{\Omega}\nabla[  \lambda_1p(M-w)
^{-\gamma}H^{p-1}H']  \nabla z\,dx\\
&  =-\lambda_1p\int_{\Omega}\Big\{  \gamma(M-w)  ^{-\gamma
-1}H^{p-1}H'\nabla w\nabla z+(M-w)  ^{-\gamma}
H^{p-1}H''\| \nabla z\| ^2  \\
&\quad  +(p-1)  (M-w)  ^{-\gamma}H^{p-2}(H')
^2\| \nabla z\| ^2\Big\} \, dx.
\end{align*}
Hence, we can write
\begin{align*}
I  &  =-\int_{\Omega}\Big\{  \lambda_2\gamma(\gamma+1) \| \nabla w\| ^2 
 +\big(\lambda_2\gamma p( M-w)  \frac{H'}{H}+\lambda_1p\gamma(M-w)
 \frac{H'}{H}\Big)  \nabla z\nabla w  \\
& \quad  +\lambda_1p\Big((p-1)  (\frac{H'}{H})  ^2
 +\frac{H''}{H}\Big)  (M-w)  ^2\| \nabla z\|
^2\}  (M-w)  ^{-\gamma-2}H^p\,dx.
\end{align*}
The formula can be rearranged in the form
\begin{equation}
I=-\int_{\Omega}(T(\nabla w,\nabla z)  )H^p(M-w)  ^{-\gamma-2}dx, \label{3.4}
\end{equation}
where
\begin{align*}
T(\nabla w,\nabla z) 
& =\gamma(\gamma+1)  \lambda _2\| \nabla w\| ^2+(\lambda_2+\lambda
_1)  p\gamma(M-w)  \frac{H'}{H}\nabla w\nabla z\\
&\quad +\lambda_1\big[  p\frac{H''}{H}+(p^2-p)  (\frac{H'}{H})  ^2\big]  (M-w)  ^2
\|\nabla z\| ^2.
\end{align*}
The discriminant $D$ of $T$ is given by
\begin{align*}
\frac{D}{(M-w)  ^2}  
&  = \gamma p[  ((
\lambda_2-\lambda_1)  ^2\gamma-4\lambda_1\lambda_2)
p+4\lambda_2\lambda_1(\gamma+1)  ]  \big(\frac{H'}{H}\big)  ^2\\
& \quad -4\lambda_2\gamma(\gamma+1)  p\frac{H''}{H}.
\end{align*}
If $\gamma$ and $p$ are chosen so as to satisfy conditions
\eqref{3.2} and \eqref{3.3}, then $D<0$ and consequently
\begin{equation}
I\leq0. \label{3.5}
\end{equation}
Now, let us examine the second part of the derivative. We have
\begin{align*}
J  &  =\int_{\Omega}(M-w)  ^{-\gamma-1}H^p(z)
\big\{  \gamma k(w,z)  +p(M-w)  \frac{H'}{H}h(w,z)  \big\}\,  dx\\
&  =\int_{\Omega}\big[ p(M-w)  (-\frac{H'}
{k})  \frac{h}{H}-\gamma\big]  (-k(w,z)  )H^p(M-w)  ^{-\gamma-1}dx.
\end{align*}
This is along to \eqref{2.9}, and \eqref{******} yields
\[
\lim_{z\to \infty}(-\frac{H'}{k})
<\frac{2\gamma}{nM}.
\]
Hence, there exists $\tilde{z}>0$ such that
\[
[  (M-w)  (-p\frac{H'}{k})-\gamma]  H^p(M-w)  ^{-\gamma-1}\leq 0,
\]
for all $z\geq\overline{z}$ and $0\leq w\leq M$. Rearranging the inequality
and simplifying yields
\[
(-\frac{H'}{k})  \leq\frac{2\gamma}{nM}\quad (0\leq w\leq M).
\]
Since the function in $J$ is continuous, it is uniformly bounded for $z\geq0$
and $0\leq w\leq M$. Therefore, there exists $C_1>0$ such that
\begin{equation}
J\leq C_1, \label{3.6}
\end{equation}
and the proof is complete.
\end{proof}


\begin{corollary} \label{coro1}
For any initial data $(u_0,v_0)$ in
$L^{\infty}(\Omega)\times L^{\infty}(\Omega)$ and any functions $f$ and
$g$ pointing into the region $\Sigma_2$ on $\partial\Sigma_2$ and satisfying
either \eqref{1.5}--\eqref{350} or \eqref{1.8}--\eqref{1.13},
the solutions of the problem \eqref{1.1}-\eqref{1.2} are global in time
and uniformly bounded on $(0,+\infty)\times\Omega$.
\end{corollary}

\begin{proof}
From \eqref{2.8}, we easily deduce that 
$h(w)\in L^{\infty}([0,T^{\ast})$, $L^p(\Omega))$ for all $p\geq1$ 
and consequently $w\in L^{\infty}([0,T^{\ast})$, $L^{\infty}(\Omega))$ 
(see \cite{Henry1981} and \cite{Haraux1983}). 
It follows that the solutions of the system
\eqref{2.1}-\eqref{2.2} are global in time and uniformly bounded on
$(0,+\infty)\times\Omega$.
\end{proof}

\section{Numerical Example}

This section will present numerical solutions for an example drawn from the
proposed model and satisfying the conditions for the existence of global solutions. 
The results are obtained through the finite difference method with
appropriately chosen descritization intervals in space and time. Let us
consider the case where $c=0$ and
\begin{equation}
\begin{gathered}
f(u,v)  =-\big(u+\frac{b}{a-d}v\big)  ^{k}e^{(\epsilon v+e^v)  }\\
g(u,v)  =\big(u+\frac{b}{a-d}v\big)  ^{l}e^{e^v},
\end{gathered}  \label{Ex1.1}
\end{equation}
with $\epsilon>1$ and $k,l>0$. Note that taking $b=0$, we obtain
\begin{gather*}
f(u,v)  =-u^{k}e^{(\epsilon v+e^v)  }\\
g(u,v)  =g(u,v)  =u^{l}e^{e^v},
\end{gather*}
which is the same example considered by Kouachi in \cite{Kouachi2010}.
However, let us consider the non-diagonal case with $b\not =0$. Consider the
diffusion matrix
\[
A=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
=\begin{pmatrix}
4 & 6\\
0 & 5
\end{pmatrix},
\]
with eigenvalues $\lambda_1=4$ and $\lambda_2=5$.
Since $c=0$, we have the invariant region $\Sigma_1$ defined as
\[
\Sigma_1=\{  (u_0,v_0)  \in\mathbb{R}^2\text{ such that }6v_0\leq u_0\}  .
\]
Let us choose
$(u_0,v_0)  =(1.5,0.15)$ and $\epsilon=3>1$, $(k,l)  =(0.6,0.8)$. The
system can now be written as
\begin{equation}
\begin{gathered}
\frac{\partial w}{\partial t}-4\Delta w=F(u,v)  =f(u,v)  -6g(u,v) \\
\frac{\partial v}{\partial t}-5\Delta v=g(u,v)  ,
\end{gathered} \label{Ex1.2}
\end{equation}
where $w=u-6v\geq0$.
We can easily verify that the resulting system satisfies conditions
\eqref{1.7.5} and \eqref{350}. Solving the resulting system \eqref{Ex1.2}
numerically yields the solutions shown in Figure \ref{Ex1_Sol} for the
one-dimensional diffusion case.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1} 
\end{center}
\caption{Solutions of \eqref{Ex1.1} in the one-dimensional case. 
The top half depicts solutions of the equivalent diagonal system $(w,v) $
 and the bottom half depicts the raw solutions $(u,v) $.}
\label{Ex1_Sol}
\end{figure}

\section{Appendix}\label{SecRemarks}

In the main result of this article, we proved the existence of global solutions
for the proposed system \eqref{1.1}--\eqref{1.3} in the region $\Sigma_2$
using an appropriate Lyapunov functional. The work can be triavially extended
to the remaining regions. Recall that we started by finding an equivalent
system in the region in question, then we used this equivalent system to prove
the existence of global solutions. The proof of Theorem \ref{TheoGlobal} in
the region $\Sigma_2$ can be extended to the remaining regions, i.e.
$\Sigma_1$, $\Sigma_{3}$, $\Sigma_{4}$, and $\Sigma_{5}$, in a similar
fashion using the following equivalent systems.
\smallskip

\noindent\textbf{Case 3.}
For $\Sigma_{3}$, a new system of two equations is formed. 
The first equation is the result of multiplying the second equation of 
\eqref{1.1} by $\mu_2$ and subtracting the first from it. The second equation 
is formed by multiplying the second equation of \eqref{1.1} by $(-\mu_1)  $
and adding the first. Hence,
\begin{equation}
\begin{gathered}
\frac{\partial(\mu_2v-u)  }{\partial t}-\Delta[  (
c\mu_2-a)  u+(d\mu_2-b)  v]  
=\mu_2 g(u,v)-f(u,v)\\
\frac{\partial(-\mu_1v+u)  }{\partial t}-\Delta[  (
a-c\mu_1)  u+(-d\mu_1+b)  v]  
=-\mu _1g(u,v)+f(u,v).
\end{gathered}  \label{4.1}
\end{equation}
Assuming without loss that $a<d$ and with the fact that $\lambda_1$ and
$\lambda_2$ are the eingenvalues of $A$, we obtain
\begin{equation}
\begin{gathered}
\frac{\partial w}{\partial t}-\lambda_2\Delta w=k(w,z) \\
\frac{\partial z}{\partial t}-\lambda_1\Delta z=h(w,z)  ,
\end{gathered} \label{4.2}
\end{equation}
where
\begin{equation}
\begin{gathered}
w=\mu_2v-u\geq0\\
z=-\mu_1v+u\geq0,
\end{gathered}  \label{4.3}
\end{equation}
and
\begin{equation}
\begin{gathered}
k(w,z)  =\mu_2g(u,v)-f(u,v)\\
h(w,z)  =-\mu_1g(u,v)+f(u,v).
\end{gathered}\label{4.4}
\end{equation}
The rest is trivial.
\smallskip

\noindent\textbf{Case 4.} 
For $\Sigma_{4}$, we form a new system of equations based on \eqref{1.1}. The
second equation is multiplied by $\mu_1$ and the first  is subtracted from
it to yield the first equation. The second is obtained by multiplying the
second equation of \eqref{1.1} by $\mu_2$ and subtracting the first from it.
Thus we obtain
\begin{equation}
\begin{gathered}
\frac{\partial(\mu_2v-u)  }{\partial t}+(a-c\mu_2)  
\Delta u+(b-d\mu_2)  \Delta v =\mu_2g(u,v)-f(u,v)\\
\frac{\partial(\mu_1v-u)  }{\partial t}+(a-c\mu_1)  
\Delta u+(b-d\mu_1)  \Delta v =\mu_1 g(u,v)-f(u,v).
\end{gathered}  \label{4.5}
\end{equation}
Then, if we assume without loss that $a<d$ and with the fact that 
$\lambda_1$ and $\lambda_2$ are the eingenvalues of $A$, we have
\begin{equation}
\begin{gathered}
\frac{\partial w}{\partial t}-\lambda_2\Delta w=k(w,z) \\
\frac{\partial z}{\partial t}-\lambda_1\Delta z=h(w,z)  ,
\end{gathered} \label{4.6}
\end{equation}
where
\begin{equation}
\begin{gathered}
w=\mu_2v-u\geq0\\
z=\mu_1v-u\geq0,
\end{gathered} \label{4.7}
\end{equation}
and
\begin{equation}
\begin{gathered}
k(w,z)  =\mu_2g(u,v)-f(u,v)\\
h(w,z)  =\mu_1g(u,v)-f(u,v).
\end{gathered}  \label{4.8}
\end{equation}
Again, the remainder of the proof is trivial.
\smallskip

\noindent\textbf{Case 5.} 
For $\Sigma_{5}$, we multiply the second equation of \eqref{1.1} by 
$(-\mu_1)  $ and add the first to it, and separately multiply it by
$(-\mu_2)  $ and add the first to it. This yields a new system
of equations:
\begin{equation}
\begin{gathered}
\frac{\partial(-\mu_2v+u)  }{\partial t}+(c\mu_2-a)  \Delta u+(d\mu_2-b) 
 \Delta v=f(u,v)-\mu_2g(u,v)\\
\frac{\partial(-\mu_1v+u)  }{\partial t}+(c\mu_1-a)  \Delta u+(d\mu_1-b) 
 \Delta v=f(u,v)-\mu_1g(u,v).
\end{gathered}  \label{4.9}
\end{equation}
Then, assuming without loss that $a<d$ and with the fact that $\lambda_1$
and $\lambda_2$ are the eingenvalues of $A$, we can write:
\begin{equation}
\begin{gathered}
\frac{\partial w}{\partial t}-\lambda_2\Delta w=k(w,z) \\
\frac{\partial z}{\partial t}-\lambda_1\Delta z=h(w,z)  ,
\end{gathered}  \label{4.10}
\end{equation}
where
\begin{equation}
\begin{gathered}
w=-\mu_2v+u\geq0\\
z=-\mu_1v+u\geq0,
\end{gathered} \label{4.11}
\end{equation}
and
\begin{equation}
\begin{gathered}
k(w,z)  =f(u,v)-\mu_2g(u,v)\\
h(w,z)  =f(u,v)-\mu_1g(u,v).
\end{gathered} \label{4.12}
\end{equation}
The rest follows in the same way as the proof for region $\Sigma_2$.


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