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\markboth{\footnotesize \emph{\emph{Global Journal of Mathematical Analysis}}}{\footnotesize \emph{\emph{Global Journal of Mathematical Analysis}}}
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\centerline {\huge{\bf Global solution of reaction diffusion}}

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\centerline{\huge{\bf system with full matrix}}

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\centerline{\bf {Abdelkader Moumeni$^{1*}$, Maroua Mebarki$^{2}$ }}

\centerline{}
{\small
\centerline{\emph{$^{1}$ Laboratoire de mathematique appliquee, dymamique et modelisation}}

\centerline{\emph{$^{2}$Badji Mokhtar university-\textbf{Annaba-Algeria}}}

%\centerline{\emph{$^{3}$Affiliation of the third author-\textbf{delete if identical with the first and second author}}}

\centerline{\emph{marwa.mebarki@hotmail.fr}}}

\centerline{}
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\newtheorem{Theorem}{Theorem}[section]

\newtheorem{Definition}[Theorem]{Definition}

\newtheorem{Corollary}[Theorem]{Corollary}

\newtheorem{Lemma}[Theorem]{Lemma}

\newtheorem{Example}[Theorem]{Example}
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\noindent \hspace{-3 pt}{\scriptsize \textbf{ Copyright \copyright 2014 Author. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.}}
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\smallskip

\noindent
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\noindent \textbf{Abstract}\\
\centerline{}
The purpose of this paper is to prove the global existence in time of
solutions for the strongly coupled reaction-diffusion system:
\begin{center}
 $\left\{
\begin{array}{l} \frac{\partial u}{\partial t}-d_{1}\Delta u-d_{2}\Delta v=f\left( u,v\right) \ \ \ \ \   in \ R^+ \times \Omega \\
\frac{\partial u}{\partial t}-d_{3}\Delta u-d_{4}\Delta v=g\left( u,v\right) \ \ \ \ \ in \ R^+ \times \Omega \\
\frac{\partial u}{\partial \eta }=\frac{\partial v}{\partial \eta }=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  in \ R^+ \times \Omega \\
u(.,0)=u_{0}(.),v(.,0)=v_{0}(.)  \ \ \ \ \ \  \ in \ \Omega 

\end{array}%
\right. $
\end{center}
with full matrix of diffusion coefficients. Our techniques of proof are
based on Lyapunov functional methods and some $L^{p}$ estimates. we show
that global solutions exist. Our investigation applied for a wide class of
the nonlinear terms$f$ and $g$.




\centerline{}
\noindent {\footnotesize \emph{\textbf{Keywords}}:  \emph{Global Existence, Reaction Diffusion Systems, Lyapunov
Functional.}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\hrulefill
%=============================
\section{Introduction}
%=============================
In this paper we study the following semilinear parabolic system

\begin{equation}
\left\{ 
\begin{array}{lllllllllll}
\dfrac{\partial u}{\partial t}-d_{1}\triangle u-d_{2}\triangle v=f(u,v) &  & 
& \bigskip & \text{in }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times \Omega &  &  &  &  &  &  \\ 
\dfrac{\partial v}{\partial t}-d_{3}\triangle u-d_{4}\triangle v=g(u,v) &  & 
& \bigskip & \text{in }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times \Omega &  &  &  &  &  & 
\end{array}%
\right.  \tag{1.1}
\end{equation}

Where $\Omega $ is a regular and bounded domain of $R^{n}$, $(n\geq
1),u=u(t,x)$

$v=v(t,x)$, $x\in \Omega ,t>0$ are real valued functions, $\Delta $denotes
the Laplacian operator, and the constants of diffusion $d_{1}$, $d_{2}$, $%
d_{3}$, $d_{4}$ are assumed to be nonnegative.

System (1.1) is subjected to the following boundary conditions 
\begin{equation}
\begin{array}{lllllllllll}
\dfrac{\partial u}{\partial \eta }=\dfrac{\partial v}{\partial \eta }=0 &  & 
& \bigskip & \text{in }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}\times \partial \Omega &  &  &  &  &  & 
\end{array}%
\   \tag{1.2}
\end{equation}

and the initial data 
\begin{equation}
\begin{array}{lllllllllll}
u(.,0)=u_{0}(.)\text{, }v(.,0)=v_{0}(.) &  &  &  & \text{in }\Omega &  &  & 
&  &  & 
\end{array}%
\   \tag{1.3}
\end{equation}

which are assumed to be nonnegative.

The above system (1.1)--(1.3) arises in physics, chemistry and various
biological processes including population dynamics. ( See [6], [23] and
references therein). condition (1.2) means that there is no species of
immegration .

Concerning the functions $f$ and $g$, we assume the following hypothesis:\ 

\textbf{(H1)} $f(r,s)$ and $g(r,s)$ are continuously differentiable on $%
R^{+}\times R^{+}$, such that%
\begin{equation}
f(0,s)\geq 0,g(r,0)\geq 0\forall r,s\geq 0\ \ \ \ \ \ \ \   \tag{1.4}
\end{equation}

(\textbf{H2}) Assume further that there exists an integer$\ \forall p\geq 1$
such that%
\begin{equation}
K^{2i-1}f\ (r,s)+g(r,s)\leq C(r+s+1)\ \ \ \ i=1,...,p\ \ \ \   \tag{1.5}
\end{equation}

$\bigskip $For all $r,s\geq 0$ \ and a real $m\geq 1$ \ sach that:%
\begin{equation}
\sup (|f(r,s)|,|g(r,s)|)\leq C(r+s+1)^{m},\forall r,s\geq 0\ \ \ \ \ \ \  
\tag{1.6}
\end{equation}

The main question we want to address is the existence of global solutions
for system (1.1)--(1.3). In fact the subject of the global existence of
reaction diffusion systems has received a lot of attention in the last
decades and several outstanding results have been proved by some of the
major experts in the field. See [3, 5, 14].

This question has been investigated by many authors by considering special
forms of the nonlinear terms $f$ and $g$.

In the trivial case where $d_{2}=d_{3}=d_{1}-d_{4}=0$; nonnegative solutions
exist globally in time.

In diagonal case where $d_{2}=$ $d_{3}=0$ Note that, Alikakos[1], treated
the following system

\begin{equation}
\left\{ 
\begin{array}{l}
u_{t}-d_{1}\triangle u=f(u,v)\ \ \ \ \ \ \ \text{\ in }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
{R}
%EndExpansion
^{+}\times \Omega \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 
v_{t}-d_{4}\triangle v=g(u,v)\ \ \ \ \ \ \ \ \text{ in }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
{R}
%EndExpansion
^{+}\times \Omega \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }%
\end{array}%
\right.  \tag{1.7}
\end{equation}

with the same boundary conditions (1.2) and initial condition (1.3), where 
\begin{equation*}
f(u,v)=-g(u,v)=-uv^{\sigma }
\end{equation*}%
and gave a positive answer to the problem of the global existence of system
(1.7), (1.2), (1.3) under the assumption 
\begin{equation}
1<\sigma <\sigma _{0}  \tag{1.8}
\end{equation}%
where 
\begin{equation}
\sigma _{0}=1+\frac{2}{n}  \tag{1.9}
\end{equation}

The method used in [1] is based on some Sobolev embedding theorems.

Note that the exponent $\sigma _{0}$ given in (1.9) is exactly the critical
exponents given by Fujita [7] for the parabolic problem

\begin{equation}
\left\{ 
\begin{array}{l}
u_{t}=\triangle u+u^{\sigma }\ \ \ \ \ \ \ \text{\ } \\ 
u\left( x,0\right) =u_{0}\left( x\right)%
\end{array}%
\right.  \tag{1.10}
\end{equation}

where $u_{0}$ in (1.10) is a nonnegative. Fujita proved that if%
\begin{equation*}
1<\sigma <\sigma _{0}
\end{equation*}%
then(1.10) possesses no global nonnegative solutions while if $\sigma
>\sigma _{0}$ , both global and nonglobal nonnegative solutions exist,
depending on the nature of the initial energy. Hollis, Martin, and Pierre
[10] established global existence of positive solutions for system
(1.1)-(1.2) with the boundary conditions%
\begin{equation*}
\lambda _{1}u+(1-\lambda _{1})\frac{\partial u}{\partial \eta }=\beta
_{1},\lambda _{2}v+(1-\lambda _{2})\frac{\partial v}{\partial \eta }=\beta
_{2}\text{on }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
{R}
%EndExpansion
^{+}\times \partial \Omega
\end{equation*}

where%
\begin{equation*}
0\prec \lambda _{1},\lambda _{2}\prec 1\text{ or }\lambda _{1}=\lambda _{2}=1%
\text{ and }\beta _{1}\succeq 0,\beta _{2}\succeq 0\text{ }
\end{equation*}

or

\begin{equation*}
\lambda _{1}=\lambda _{2}=\beta _{1}=\beta _{2}=0
\end{equation*}

under the conditions $f(r,s)+g(r,s)\leq C(r,s)(r+s+1);\forall r,s\succeq
0,i=1,...,p$

In [20] Masuda obtained a global existence result for a large class of the
parameter $\sigma $. In fact, by using some $L^{p}$ estimates, he showed
that the solution of problem (1.1)--(1.3) exists globally in time if $\sigma
>1$.

The same result in [20] was obtained by Hollis et al [19] by exploiting the
duality arguments on $L^{p}$ techniques, allowing to derive the uniform
boundeness of the solution.

Following Masuda's approach, Haraux and Youkana [9] established a global
existence result of system (1.1)--(1.3) for a large class of the function $f$
and $g$. More precisely they showed that for%
\begin{equation}
f(u,v)=-g(u,v)=-u\Phi (v)\ \   \tag{1.11}
\end{equation}

the problem (1.1)--(1.3) admits a global solution provided that the
following condition holds:

\begin{equation*}
\lim_{\left( v\rightarrow +\infty \right) }\frac{\log \left( 1+\Phi \left(
v\right) \right) }{v}=0
\end{equation*}

In the general case, that is to say for%
\begin{equation}
f(u,v)=-g(u,v)\ \   \tag{1.12}
\end{equation}

the positivity of the function $g(u,v)$ together with the maximum principle
of the heat operator give the following uniform estimate of the solution in $%
L^{\infty }(\Omega )$ 
\begin{equation*}
\Vert u(t)\Vert _{\infty }\leq \Vert u_{0}(t)\Vert _{\infty }\forall t\in
\lbrack 0,T_{max}[
\end{equation*}%
Where $T_{max}$ is the maximal time of existence. See Pazy [24] for more
details.Based on the Lyapunov functional method and for $f$ and $g$
satisfying (1.12), Kouachi [12] proved that the solution of problem
(1.1)--(1.3) exists globally in time if 
\begin{equation*}
\lim_{\left( v\rightarrow +\infty \right) }\frac{\log (1+f(u,v))}{v}\prec 
\frac{8\alpha \beta }{n(\alpha -\beta )^{2}\Vert u_{0}\Vert _{\infty }}
\end{equation*}

Moumeni and Salah Derradji [21] have established the existence of global
solution using an approach that involves the Lyapunov's functional for the
system (1.1)--(1.3) where the functions $f$ and $g$ are assumed to satisfy
the condition $f\ (r,s)+g(r,s)\leq C(r+s+1)$ .

If $d_{1}\neq d_{4}$ , an important particular case is that when $f\leq 0$,
which means that the first substance is absorbed by the reaction, in this
case, the problem of the global existence of system (1.7) reduces to
obtaining a uniform estimate for $v$, since by the maximal principle we have 
$u(x,t)\leq \Vert u_{0}\Vert _{\infty }$.

Here the global existence when $d_{1}\succ d_{4}$ has been treated by Kanel
and Kirane [12] for a bounded domain and by Martin and Pierre [14] for whole
space $R^{n}$.

Still in the case $d_{1}\neq d_{4}$ , but without assuming $d_{1}\succ d_{4}$
, the answer is again positive to the problem of the global existence of
system (1.7) under condition (1.13) and a polynomial growth assumption on $g$%
:

$g(u,v)\leq C(u+v+1)^{\gamma }$,for all$u,v\geq 0$ and some $\gamma \geq 1$%
,see [10] for more details.

If the diffusion coefficients are the same, that is, if $d_{1}=d_{4}$, then
system (1.7) has a global solution under the condition 
\begin{equation}
f(u,v)+g(u,v)\leq 0  \tag{1.13}
\end{equation}%
,which is known as the mass dissipative structure condition. Indeed if

$d_{1}=d_{4}$,then the solution $(u,v)$ of (1.7) satisfies (by summing up
the two equations in (1.7))

\begin{equation*}
\frac{\partial (u+v)}{\partial t}-d_{1}(u+v)=f+g\leq 0
\end{equation*}%
Then the maximal principle implies%
\begin{equation*}
0\leq u+v\leq \Vert u_{0}\Vert _{\infty }+\Vert v_{0}\Vert _{\infty }
\end{equation*}

Therefore, the global existence follows.

In tridiagonal case where $d_{3}>0$ and $d_{2}=0$, Moumeni and Salah
Derradji [22] have established the existence of global solution of the
problem (1.1)--(1.3) using the Lyapunov method combined with some $L^{p}$
estimates.

For $d_{3}>0$ and $d_{2}\succ 0$ In [12] J. I. Kanel and M. Kirane proved
the global existence of solutions for a strongly coupled reaction-diffusion
system with homogeneous Neumann boundary conditions and%
\begin{equation*}
f(u,v)=-g(u,v)=uv^{m},m\succ 0
\end{equation*}

$m$ is an odd integer, Later they improved their results in [13] where they
obtained the global existence with 
\begin{equation*}
f(u,v)=-g(u,v)=u{F}(v)
\end{equation*}

On the same direction, S. Kouachi [17] has proved the global existence of
solutions for two-component reaction-diffusion systems with a general full
matrix of diffusion coefficients, nonhomogeneous boundary conditions and
polynomial growth conditions on the nonlinear terms and he obtained in [18]
the global existence of solutions for the same system with homogeneous
Neumann boundary conditions and 
\begin{equation*}
g(u,v)=\rho F(u,v),f(u,v)=-\sigma F(u,v)\rho \succ 0\ ,\sigma \succ 0
\end{equation*}

B. Rebiai and S. Benachour[25]treat the case of a general full matrix of
diffusion coefficients with the homogeneous boundary conditions with
nonlinearities of exponentiel growth .

finally in[4] K. Boukerrioua generalize a result obtained in [22]. Our
techniques are based on invariant regions and Lyapunov functional methods.

In the present work we consider problem (1.1)--(1.3) with $d_{2}>0$ and $%
d_{3}>0$ , where the function $f$ and $g$ are assumed to satisfy the
condition (1.6), and by adopting the Lyapunov method combined with some $%
L^{p}$ estimates we establish a global existence result of the solution .

The content of this paper is as follows. In section 2, we introduce some
notations and give a local existence result. Our main result is stated in
section 3.

\section{\textbf{Local existence}}

Throughout this work, we denote by$\left\Vert .\right\Vert _{p}$, $p\in
\lbrack 1;+\infty )$the norm in $L^{p}$ and $\left\Vert .\right\Vert
_{\infty }$ the norm in $C(\overline{\Omega })$ or $L^{\infty }$,
respectively, defined by $\left\Vert u\right\Vert _{p}=\int_{\Omega
}\left\vert u\right\vert ^{p}dx^{\frac{1}{p}}$ \ and $\left\Vert
u\right\Vert _{\infty }=ess\underset{x\in \Omega }{\sup }\left\vert
u(x)\right\vert $

The study of local existence and uniqueness of solutions $(u;v)$ of
(1.1)-(1.3) follows from the basic existence theory for parabolic semi
linear equations (see, e.g., [2], [10], [24] and [27]). As a consequence,
for any initial data in $L^{\infty }$there exists a $T_{\max }\in (0;+\infty
]$ such that (1.1)-(1.3) has a unique classical solution on $\left(
0,T_{\max }\right[ \times \Omega $ . Furthermore,

if $T_{\max }\prec \infty $ then $\lim_{t\rightarrow T_{\max }}\left\{
\left\Vert u(t,.)\right\Vert _{\infty },\left\Vert v(t,.)\right\Vert
_{\infty }\right\} =+\infty $

Therefore, if there exists a positive constant $C$ such that

$\left\Vert u(t,.)\right\Vert _{\infty }+\left\Vert v(t,.)\right\Vert
_{\infty }\preceq C$ $\forall t\in \lbrack 0,T_{\max })$ then $T_{\max
}=+\infty $

\textbf{Remark2.1}

Under condition (\textbf{H1}), it follows from the invariant region method
that system (1.1)--(1.3) preserves positivity. In other words, if the
initial data $u_{0}$ and $v_{0}$ in (1.3) are nonnegative, then the
functions $u$ and $v$ of the corresponding solution of (1.1)--(1.3) are also
nonnegative on $]0,T_{\max }[\times \Omega $. See [10].

\section{\textbf{Statement of the main results}}

\subsection{\textbf{Existence of global solutions}}

In this section, we state and prove our global existence result of system
(1.1)--(1.3). Our main theorem reads as follows.

\textbf{Theorem3.1}

\bigskip Let $p\succ \frac{mn}{2}.$ Assume that condition (H2) are
satisfied. Then the solution $(u(t,.),v(t,.))$ of (1.1)--(1.3) with initial
positive condition in $L^{\infty }\left( \Omega \right) $ exists globally in
time.

\bigskip

We note that to prove Theorem 3.1 it is sufficient to derive a uniform
estimate of $\sup (\left\Vert f(u,v)\right\Vert _{q},\left\Vert
g(u,v)\right\Vert _{q})$ for some $q>n/2$. (See [10] for more details).

The following lemma is a useful tool in the proof of the Theorem 3.1.

\textbf{Lemma3.1}

Let $\left( u(t,.),v(t,.)\right) $ be the solution of (1.1)--(1.3) and let $%
L(t)=\int_{\Omega }\sum_{i=0}^{p}C_{p}^{i}K^{i^{2}}u^{i}v^{p-i}dx$ wih $p$ a
positive integer

and $K$ is a serie of positive numbers such that\ $K\succeq \max (\frac{%
d_{1}+d_{4}}{2\sqrt[1]{d_{1}d_{4}}},\frac{d_{2}+d_{3}}{2\sqrt[1]{d_{3}d_{2}}}%
)$

then the functional $L$ is uniformly bounded on the interval $\left[
0,T^{\ast }\right] $ \ $T^{\ast }\preceq T_{\max }$

\textbf{Proof}

Differentiating $L$ with respect to t yields

\begin{eqnarray*}
L^{\prime }(t) &=&\int_{\Omega }\left[
\sum_{i=1}^{p}(iC_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i})u_{t}+%
\sum_{i=0}^{p-1}((p-i)C_{p}^{i}K^{i^{2}}u^{i}v^{p-i-1})v_{t}\right] dx \\
&=&\int_{\Omega
}\sum_{i=1}^{p}(iC_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i})(d_{1}\triangle
u+d_{2}\triangle v+f(u,v))dx+ \\
&&\int_{\Omega
}\sum_{i=0}^{p-1}((p-i)C_{p}^{i}K^{i^{2}}u^{i}v^{p-i-1})(d_{3}\triangle
u+d_{4}\triangle v+g(u,v))dx
\end{eqnarray*}

\bigskip A simple computation leads%
\begin{eqnarray*}
L^{\prime }(t) &=&\int_{\Omega
}\sum_{i=1}^{p}(iC_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i})(d_{1}\triangle
u+d_{2}\triangle v+f(u,v))dx+ \\
&&\int_{\Omega
}\sum_{i=1}^{p}((p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i})(d_{3}%
\triangle u+d_{4}\triangle v+g(u,v))dx
\end{eqnarray*}

From the above equality, it follows that

\begin{eqnarray*}
L^{\prime }(t) &=&\int_{\Omega
}\sum_{i=1}^{p}d_{1}iC_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i}\triangle
udx+\int_{\Omega
}\sum_{i=1}^{p}d_{4}(p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i}\triangle
vdx \\
&&+\int_{\Omega
}\sum_{i=1}^{p}d_{2}iC_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i}\triangle
vdx+\int_{\Omega
}\sum_{i=1}^{p}d_{3}(p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i}\triangle
udx \\
&&+\int_{\Omega
}\sum_{i=1}^{p}iC_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i}f(u,v)dx+\int_{\Omega
}\sum_{i=1}^{p}(p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i}g(u,v)dx \\
&&I+J+H
\end{eqnarray*}

By a simple use of Green's formula we have:

\begin{equation}
I=-\int_{\Omega }\left( A\left\vert \nabla u\right\vert ^{2}+B\nabla u\nabla
v+C\left\vert \nabla v\right\vert ^{2}\right) dx  \tag{3.1}
\end{equation}

where:%
\begin{equation*}
A=\sum_{i=2}^{p}d_{1}i\left( i-1\right) C_{p}^{i}K^{i^{2}}u^{i-2}v^{p-i}
\end{equation*}

\bigskip 
\begin{equation*}
B=\sum_{i=1}^{p-1}d_{1}i\left( p-i\right)
C_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i-1}+\sum_{i=2}^{p}d_{4}\left( i-1\right)
(p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-2}v^{p-i}
\end{equation*}

\begin{equation*}
C=\sum_{i=1}^{p-1}d_{4}\left( p-i\right)
(p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i-1}
\end{equation*}

Using the fact that :%
\begin{equation}
iC_{p}^{i}=(p-i+1)C_{p}^{i-1}=pC_{p-1}^{i-1}\text{ \ \ \ }\forall i=1,...,p 
\tag{3.2}
\end{equation}

and also since 
\begin{equation}
i(i-1)C_{p}^{i+1}=i(p-i)C_{p}^{i}=\left( p-i\right)
(p-i+1)C_{p}^{i-1}=p(p-1)C_{p-2}^{i-2}  \tag{3.3}
\end{equation}

we get 
\begin{equation*}
A=\sum_{i=2}^{p}d_{1}p\left( p-1\right) C_{p-2}^{i-2}K^{i^{2}}u^{i-2}v^{p-i}
\end{equation*}

\begin{eqnarray*}
B &=&\sum_{i=1}^{p-1}d_{1}p\left( p-1\right)
C_{p-2}^{i-2}K^{i^{2}}u^{i-1}v^{p-i-1}+%
\sum_{i=2}^{p}d_{4}p(p-1)C_{p-2}^{i-2}K^{(i-1)^{2}}u^{i-2}v^{p-i} \\
&=&B_{1}+B_{2}
\end{eqnarray*}

and 
\begin{equation*}
C=\sum_{i=1}^{p-1}d_{4}p(p-1)C_{p-2}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i-1}
\end{equation*}

Putting :$j=i-2$ ,we have :

\begin{equation*}
A=\sum_{j=0}^{p-2}d_{1}p\left( p-1\right)
C_{p-2}^{j}K^{(j+2)^{2}}u^{j}v^{p-j-2}
\end{equation*}

\begin{equation*}
B_{2}=\sum_{j=0}^{p-2}d_{4}p(p-1)C_{p-2}^{j}K^{(j+1)^{2}}u^{j}v^{p-j-2}
\end{equation*}

and Putting :$j=i-1$ ,we get :%
\begin{equation*}
B_{1}=\sum_{j=0}^{p-2}d_{1}p\left( p-1\right)
C_{p-2}^{j}K^{(j+1)^{2}}u^{j}v^{p-j-2}
\end{equation*}

\begin{equation*}
C=\sum_{j=0}^{p-2}d_{4}p(p-1)C_{p-2}^{j}K^{j^{2}}u^{j}v^{p-j-2}
\end{equation*}

Then :%
\begin{equation}
I=-p(p-1)\sum_{j=0}^{p-2}C_{p-2}^{j}\int_{\Omega }u^{j}v^{p-j-2}\times \Psi
(\nabla u,\nabla v)dx  \tag{3.4}
\end{equation}

where%
\begin{equation*}
\Psi (\nabla u,\nabla v)=d_{1}K^{(j+2)^{2}}\left\vert \nabla u\right\vert
^{2}+(d_{1}+d_{4})K^{(j+1)^{2}}\nabla u\nabla v+d_{4}K^{j^{2}}\left\vert
\nabla v\right\vert ^{2}
\end{equation*}

The quadratic forms are positive since :%
\begin{equation}
((d_{1}+d_{4})K^{(j+1)^{2}})^{2}-4d_{1}d_{4}K^{j^{2}}K^{(j+2)^{2}}\preceq 0%
\text{ \ \ }j=0,...,p-2  \tag{3.5}
\end{equation}%
$\ $Using the relation $K\succeq \max (\frac{d_{1}+d_{4}}{2\sqrt[1]{%
d_{1}d_{4}}},\frac{d_{2}+d_{3}}{2\sqrt[1]{d_{3}d_{2}}})$

Then 
\begin{equation}
I\preceq 0  \tag{3.6}
\end{equation}

By a simple use of Green's formula we have:

\begin{equation}
J=-\int_{\Omega }\left( D\left\vert \nabla v\right\vert ^{2}+E\nabla v\nabla
u+F\left\vert \nabla u\right\vert ^{2}\right) dx  \tag{3.7}
\end{equation}

where:%
\begin{equation*}
D=\sum_{i=1}^{p-1}d_{2}i\left( p-i\right) C_{p}^{i}K^{i^{2}}u^{i-1}v^{p-i-1}
\end{equation*}

\begin{equation*}
E=\sum_{i=2}^{p}d_{2}i\left( i-1\right)
C_{p}^{i}K^{i^{2}}u^{i-2}v^{p-i}+\sum_{i=1}^{p-1}d_{3}\left( p-i\right)
(p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i-1}
\end{equation*}

\begin{equation*}
F=\sum_{i=2}^{p}d_{3}\left( i-1\right)
(p-i+1)C_{p}^{i-1}K^{(i-1)^{2}}u^{i-2}v^{p-i}
\end{equation*}

Using the relation (3.2) we get 
\begin{equation*}
D=\sum_{i=1}^{p-1}d_{2}p\left( p-1\right)
C_{p-2}^{i-2}K^{i^{2}}u^{i-1}v^{p-i-1}
\end{equation*}

\begin{eqnarray*}
E &=&\sum_{i=2}^{p}d_{2}p\left( p-1\right)
C_{p-2}^{i-2}K^{i^{2}}u^{i-2}v^{p-i}+%
\sum_{i=1}^{p-1}d_{3}p(p-1)C_{p-2}^{i-1}K^{(i-1)^{2}}u^{i-1}v^{p-i-1} \\
&&E_{1}+E_{2}
\end{eqnarray*}

and 
\begin{equation*}
F=\sum_{i=2}^{p}d_{3}p(p-1)C_{p-2}^{i-2}K^{(i-1)^{2}}u^{i-2}v^{p-i}
\end{equation*}

putting :$j=i-1$ ,we have :%
\begin{equation*}
D=\sum_{j=0}^{p-2}d_{2}p\left( p-1\right)
C_{p-2}^{j}K^{(j+1)^{2}}u^{j}v^{p-j-2}
\end{equation*}

\begin{equation*}
E_{2}=\sum_{j=0}^{p-2}d_{3}p(p-1)C_{p-2}^{j}K^{j^{2}}u^{j}v^{p-j-2}
\end{equation*}

and putting :$j=i-2$ ,we get :

\begin{equation*}
E_{1}=\sum_{j=0}^{p-2}d_{2}p\left( p-1\right)
C_{p-2}^{j}K^{(j+2)^{2}}u^{j}v^{p-j-2}
\end{equation*}

\begin{equation*}
F=\sum_{j=0}^{p-2}d_{3}p(p-1)C_{p-2}^{j}K^{(j+1)^{2}}u^{j}v^{p-j-2}
\end{equation*}

Then :%
\begin{equation}
J=-p(p-1)\sum_{j=0}^{p-2}C_{p-2}^{j}\int_{\Omega }u^{j}v^{p-j-2}\times \Phi
\left( \nabla v,\nabla u\right) dx  \tag{3.8}
\end{equation}

where 
\begin{equation*}
\Phi \left( \nabla v,\nabla u\right) =d_{2}K^{(j+1)^{2}}\left\vert \nabla
v\right\vert ^{2}+(d_{2}K^{(j+2)^{2}}+d_{3}K^{j^{2}})\nabla v\nabla
u+d_{3}K^{(j+1)^{2}}\left\vert \nabla u\right\vert ^{2}
\end{equation*}

The quadratic forms are positive since :%
\begin{equation}
((d_{2}K^{(j+2)^{2}}+d_{3}K^{j^{2}}))^{2}-4d_{2}d_{3}K^{(j+1)^{2}}K^{(j+1)^{2}}\preceq 0%
\text{ \ \ }j=0,...,p-2  \tag{3.9}
\end{equation}

Using the relation $K\succeq \max (\frac{d_{1}+d_{4}}{2\sqrt[1]{d_{1}d_{4}}}$
$,\frac{d_{2}+d_{3}}{2\sqrt[1]{d_{3}d_{2}}})$

Then 
\begin{equation}
J\preceq 0  \tag{3.10}
\end{equation}

Using the relation (3.2), in the third integral, yields :

\begin{equation*}
H=\int_{\Omega }\left[ p\sum_{i=1}^{p}\left( K^{i^{2}}f(u,v)+K^{\left(
i-1\right) ^{2}}g(u,v)\right) C_{p-1}^{i-1}u^{i-1}v^{p-i}\right] dx
\end{equation*}

Using the relation(1.5) we deduce 
\begin{equation*}
H\preceq c_{3}\int_{\Omega }\left[
\sum_{i=1}^{p}(u+v+1)C_{p-1}^{i-1}u^{i-1}v^{p-i}\right] dx
\end{equation*}

To prove that the functional $L$ is uniformly bounded on the interval$\left[
0,T^{\ast }\right] $

first we write

\begin{equation*}
L^{\prime }(t)\preceq c_{3}\int_{\Omega }\left[
\sum_{i=1}^{p}C_{p-1}^{i-1}u^{i}v^{p-i}+%
\sum_{i=1}^{p}C_{p-1}^{i-1}u^{i-1}v^{p-i+1}+%
\sum_{i=1}^{p}C_{p-1}^{i-1}u^{i-1}v^{p-i}\right] dx
\end{equation*}

\begin{equation*}
L^{\prime }(t)\preceq c_{3}\int_{\Omega }\left[
\sum_{i=1}^{p}C_{p-1}^{i-1}u^{i}v^{p-i}+%
\sum_{i=0}^{p-1}C_{p-1}^{i}u^{i}v^{p-i}+%
\sum_{i=0}^{p-1}C_{p-1}^{i}u^{i}v^{p-i-1}\right] dx
\end{equation*}

\begin{equation*}
L^{\prime }(t)\preceq c_{3}\int_{\Omega }\left[
\sum_{i=0}^{p}C_{p}^{i}u^{i}v^{p-i}+\sum_{i=0}^{p-1}C_{p-1}^{i}u^{i}v^{p-i-1}%
\right] dx
\end{equation*}

Using the fact that 
\begin{equation*}
\sum_{i=0}^{p-1}C_{p-1}^{i}u^{i}v^{p-i-1}=\left( u+v\right) ^{p-1}
\end{equation*}

Therefore, the last inequality can be written as 
\begin{equation*}
L^{\prime }(t)\preceq c_{1}(p)L(t)+c_{3}\int_{\Omega }\left( u+v\right)
^{p-1}
\end{equation*}

Applying H\^{o}lder's inequality to the second term in the right hand side of

the above inequality, we obtain

\begin{equation*}
L^{\prime }(t)\preceq c_{1}(p)L(t)+c_{3}(mes\Omega )^{\frac{1}{p}%
}(\int_{\Omega }\left( u+v\right) ^{p}dx)^{\frac{(p-1)}{p}})
\end{equation*}

Since the following inequality holds,

\begin{equation*}
\left( u+v\right) ^{p}=\sum_{i=0}^{p}C_{p}^{i}u^{i}v^{p-i}\preceq \frac{%
\sup_{0\preceq i\preceq p}C_{p}^{i}}{\min_{0\preceq i\preceq
p}C_{p}^{i}K^{i^{2}}}\sum_{i=0}^{p}C_{p}^{i}K^{i^{2}}u^{i}v^{p-i}
\end{equation*}

Then, we have 
\begin{equation*}
L^{\prime }(t)\preceq c_{1}(p)L(t)+c_{3}(mes\Omega )^{\frac{1}{p}}(\frac{%
\sup_{0\preceq i\preceq p}C_{p}^{i}}{\min_{0\preceq i\preceq
p}C_{p}^{i}K^{i^{2}}})^{\frac{(p-1)}{p}}(L(t))^{\frac{(p-1)}{p}}\text{ \ \ \
\ \ \ \ \ \ }\forall t\prec T_{\max }
\end{equation*}

Hence,$L(t)$ the functional satisfies the following differential inequality:%
\begin{equation*}
L^{\prime }(t)\preceq c_{1}(p)L(t)+c_{2}(p)(L(t))^{\frac{(p-1)}{p}}\text{ \
\ \ \ \ \ \ \ \ }\forall t\prec T_{\max }
\end{equation*}

where 
\begin{equation*}
c_{2}(p)=c_{3}(mes\Omega )^{\frac{1}{p}}(\frac{\sup_{0\preceq i\preceq
p}C_{p}^{i}}{\min_{0\preceq i\preceq p}C_{p}^{i}K^{i^{2}}})^{\frac{(p-1)}{p}}
\end{equation*}

which gives us, by a simple integration 
\begin{equation}
(L(t))^{\frac{1}{p}}\preceq \left[ (L(0))^{\frac{1}{p}}+\frac{c_{2}^{\prime
}(p)}{c_{1}^{\prime }(p)}\right] \exp (c_{1}^{\prime }(p)t)-\frac{%
c_{2}^{\prime }(p)}{c_{1}^{\prime }(p)}  \tag{3.11}
\end{equation}

where%
\begin{equation*}
c_{1}^{\prime }(p)=\frac{c_{1}(p)}{p}\text{ \ \ \ \ \ \ \ \ \ \ }%
c_{2}^{\prime }(p)=\frac{c_{2}(p)}{p}\text{\ }
\end{equation*}

By using the inequality 
\begin{equation*}
L(t)=\int_{\Omega }(\sum_{i=0}^{p}C_{p}^{i}K^{i^{2}}u^{i}v^{p-i})dx\succeq
\int_{\Omega }(C_{p}^{p}K^{p^{2}}u^{p}+C_{p}^{0}K^{0^{2}}v^{p})dx\text{ }
\end{equation*}

it follows that 
\begin{equation*}
L(t)\succeq \min (C_{p}^{0}K^{0^{2}},C_{p}^{p}K^{p^{2}})\sup (\int_{\Omega
}u^{p}dx,\int_{\Omega }v^{p}dx)\text{ }
\end{equation*}

Hence, 
\begin{equation*}
(L(t))^{\frac{1}{p}}\succeq \lbrack \min
(C_{p}^{0}K^{0^{2}},C_{p}^{p}K^{p^{2}})]^{\frac{1}{p}}\sup ((\int_{\Omega
}u^{p}dx)^{\frac{1}{p}},(\int_{\Omega }v^{p}dx)^{\frac{1}{p}})
\end{equation*}

And therefore,

\begin{equation}
\sup (\left\Vert u(t,.)\right\Vert _{p},\left\Vert v(t,.)\right\Vert
_{p})\preceq \frac{(L(t))^{\frac{1}{p}}}{[\min
(C_{p}^{0}K^{0^{2}},C_{p}^{p}K^{p^{2}})]^{\frac{1}{p}}}\text{ \ \ }\forall
t\prec T_{\max }  \tag{3.12}
\end{equation}

With (3.11) and (3.12) we obtain :

\begin{equation}
\sup (\left\Vert u(t,.)\right\Vert _{p},\left\Vert v(t,.)\right\Vert
_{p})\preceq c(t)\text{ \ \ \ \ }\forall t\prec T_{\max }  \tag{3.13}
\end{equation}

where%
\begin{equation*}
c(t)=\frac{1}{[\min (C_{p}^{0}K^{0^{2}},C_{p}^{p}K^{p^{2}})]^{\frac{1}{p}}}\{%
\left[ (L(0))^{\frac{1}{p}}+\frac{c_{2}^{\prime }(p)}{c_{1}^{\prime }(p)}%
\right] \exp (c_{1}^{\prime }(p)t)-\frac{c_{2}^{\prime }(p)}{c_{1}^{\prime
}(p)}\}
\end{equation*}

The proof of Lemma 3.1 is complete.

\textbf{Proof of theorem3.1 }

From (1.6)we have 
\begin{equation*}
\sup (\left\vert f(u,v)\right\vert ,\left\vert g(u,v)\right\vert )\preceq
c_{2}\left( u+v+1\right) ^{m}
\end{equation*}

Then, it follows that 
\begin{equation*}
\sup (\int_{\Omega }\left\vert f(u,v)\right\vert ^{\frac{p}{m}%
}dx,\int_{\Omega }\left\vert g(u,v)\right\vert ^{\frac{p}{m}}dx\preceq
c_{2}^{\frac{p}{m}}\int_{\Omega }\left( u+v+1\right) ^{p}dx
\end{equation*}

which implies :%
\begin{equation}
\sup (\left\Vert f(u,v)\right\Vert _{\frac{p}{m}}^{\frac{p}{m}},\left\Vert
g(u,v)\right\Vert _{\frac{p}{m}}^{\frac{p}{m}})\preceq c_{2}^{\frac{p}{m}%
}\int_{\Omega }\left( u+v+1\right) ^{p}dx  \tag{3.14}
\end{equation}

On the other hand, we have 
\begin{equation*}
\int_{\Omega }\left( u+v+1\right) ^{p}dx=\int_{\Omega
}^{k}\sum_{k=0}^{p}C_{p}^{k}\left( u+v\right) ^{k}dx
\end{equation*}

\begin{equation*}
\int_{\Omega }\left( u+v+1\right) ^{p}dx=\int_{\Omega }[1+\left( u+v\right)
^{p}]dx+\sum_{k=1}^{p-1}C_{p}^{k}\int_{\Omega }\left( u+v\right) ^{k}
\end{equation*}

An application of H\^{o}lder's inequality leads 
\begin{equation*}
\sum_{k=1}^{p-1}C_{p}^{k}\int_{\Omega }\left( u+v\right) ^{k}\preceq
\sum_{k=1}^{p-1}C_{p}^{k}\left[ \int_{\Omega }\left( 1^{\frac{p}{(p-k)}%
}dx\right) ^{\frac{(p-k)}{p}}\left( \int_{\Omega }(u+v)^{p}dx\right) ^{\frac{%
k}{p}}\right]
\end{equation*}

Hence 
\begin{eqnarray}
\int_{\Omega }\left( u+v+1\right) ^{p}dx &\preceq &mes(\Omega )+\int_{\Omega}\left( u+v\right) ^{p}dx  \TCItag{3.15} \\
&&+\sum_{k=1}^{p-1}C_{p}^{k}\left[ \left( mes(\Omega )\right) ^{\frac{(p-k)}{%
p}}\left( \int_{\Omega }\left( u+v\right) ^{p}dx\right) ^{\frac{k}{p}}\right]
\notag
\end{eqnarray}

using (3.13) we get:

\begin{equation*}
(\int_{\Omega }\left( u+v\right) ^{p}dx)^{\frac{1}{p}}=\left\Vert
u(t,.)+v(t,.)\right\Vert _{p}\preceq \left\Vert u(t,.)\right\Vert
_{p}+\left\Vert v(t,.)\right\Vert _{p}\preceq 2c(t)
\end{equation*}

and the inequality (3.15) can be written as follows 
\begin{equation*}
\int_{\Omega }\left( u+v+1\right) ^{p}dx\preceq mes(\Omega
)+2^{p}(c(t))^{p}+\sum_{k=1}^{p-1}C_{p}^{k}[(mes(\Omega ))^{\frac{(p-k)}{p}%
}(2c(t))^{k}
\end{equation*}

\begin{equation*}
\preceq \sum_{k=0}^{p}C_{p}^{k}[(mes(\Omega ))^{\frac{(p-k)}{p}}(2c(t))^{k}
\end{equation*}

Therefore 
\begin{equation}
\sup ((\left\Vert f(u,v)\right\Vert _{\frac{p}{m}}^{\frac{p}{m}},\left\Vert
g(u,v)\right\Vert _{\frac{p}{m}}^{\frac{p}{m}})\preceq c^{\frac{p}{m}%
}\sum_{k=0}^{p}C_{p}^{k}[(mes(\Omega ))^{\frac{(p-k)}{p}}(2c(t))^{k} 
\tag{3.16}
\end{equation}

which gives that

\begin{equation}
\sup (\left\Vert f(u,v)\right\Vert _{\frac{p}{m}},\left\Vert
g(u,v)\right\Vert _{\frac{p}{m}})\preceq c_{p,m}(t)\text{ \ \ }\forall
t\prec T_{\max }  \tag{3.17}
\end{equation}

where

\begin{equation*}
c_{p,m}(t)=c[\sum_{k=0}^{p}2^{k}C_{p}^{k}[(mes(\Omega ))^{\frac{(p-k)}{p}%
}(c(t))^{k}]^{\frac{p}{m}}
\end{equation*}

\textbf{Remark3.1}

From both Lemma 3.1 and Theorem 3.1, we have obtained an uniform estimate of 
$\sup (\left\Vert f(u,v)\right\Vert _{q},\left\Vert g(u,v)\right\Vert _{q})$

with $q=p/m>n/2$. By the preliminary remarks, we conclude that the solution
of the given problem exists globally in time.


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\end{document}
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