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\markboth{\hfil Existence of global solutions for systems  \hfil 
EJDE--2002/74}
{EJDE--2002/74\hfil Salah Badraoui\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 74, pp. 1--10. \newline
ISSN: 1072-6691. URL: 
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
\vspace{\bigskipamount} \\
%
  Existence of global solutions for systems of reaction-diffusion
  equations on unbounded domains
%
\thanks{ {\em Mathematics Subject Classifications:} 35B40, 35B45, 35K55, 35K65.
\hfil\break\indent
{\em Key words:} 
Reaction-diffusion systems, positivity, global existence, boundedness.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted December 5, 2001. Published August 19, 2002.} }
\date{}
%
\author{Salah Badraoui}
\maketitle

\begin{abstract}
  We consider, an initial-value problem for the thermal-diffusive
  combustion system
  \begin{gather*}
  u_t=a\Delta u-uh(v) \\
  v_t=b\Delta u+d\Delta v+uh(v),
  \end{gather*}
  where $a>0$, $d>0$, $b\neq 0$, $x\in \mathbb{R}^n$, $n\geq 1$,
  with $h(v)=v^m$, $m$ is an even nonnegative integer, and the
  initial data $u_0$, $v_0$ are bounded uniformly continuous and
  nonnegative.  It is known that by a simple comparison if
  $b=0$, $a=1$, $d\leq 1$ and $h(v)=v^m$ with $m\in \mathbb{N}^*$,
  the solutions are uniformly bounded in time.
  When $d>a=1$, $b=0$, $h(v)=v^m$ with $m\in \mathbb{N}^*$,
  Collet and Xin \cite{c1} proved the existence of global classical
  solutions and showed that the $L^\infty $ norm of $v$ can not
  grow faster than  $O(\log\log t)$ for any space dimension.
  In our case, no comparison principle seems to apply.
  Nevertheless using techniques form \cite{c1}, we essentially prove
  the existence of global classical solutions if  $a<d$, $b<0$,
  and  $v_0\geq \frac b{a-d}u_0$.
\end{abstract}


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

\section{Introduction}

In this paper, we are concerned with the existence of global solutions of 
the reaction-diffusion system
\begin{equation} \label{e1.1}
\begin{gathered}
u_t=a\Delta u-uv^m$,\quad  ($x,t)\in \mathbb{R}^n\times (0,\infty), \\
v_t=b\Delta u+d\Delta v+uv^m$,\quad  ($x,t)\in \mathbb{R}^n\times
(0,\infty ),\end{gathered}
\end{equation}
with initial data
\begin{equation} \label{e1.2}
u(x,0)=u_0(x), \quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^n.
\end{equation}
In \eqref{e1.1}, the constants $a$, $b$, $d$ are such that $a>0$, $d>0$, 
$b\neq 0$, $m$ is an even nonnegative integer.
Also $4ad\geq b^2$ which reflects the parabolicity of the system.
$\Delta $ is the Laplace operator in $x$.
In \eqref{e1.2}, the initial data $u_0$, $v_0$ are nonnegative and are in
$C_{UB}(\mathbb{R}^n)$ the space of uniformly bounded continuous functions
on $\mathbb{R}^n$.

One of the basic questions for \eqref{e1.1} with $L^\infty $ initial data is 
the
Existence of global solutions and possibly bounds uniform in time.
For $b=0$ and the Arrhenius reaction, i.e. with
$u\exp \left\{-E/v\right\}$, $E>0$ replacing $uv^m$ in \eqref{e1.1},
there are many works on global solutions, see Avrin \cite{a1}, Larrouturou
\cite{l1}
for results in one space dimension, among others. Recently, Collet and Xin
\cite{c1}
has studied the system \eqref{e1.1} but with $b=0$; they proved the
existence of global solutions and showed that the $L^\infty $ norm of $v$
can not grow faster than $O(\log\log t)$ for any space dimension.

The system
\begin{gather*}
u_t=a\Delta u-uh(v), \quad (x,t)\in \Omega \times (0,\infty )\\
v_t=b\Delta u+d\Delta v+uh(v), \quad (x,t)\in \Omega \times (0,\infty ),
\end{gather*}
on a bounded domain $\Omega \subset \mathbb{R}^n$ with Neumann boundary
conditions, $b\geq 0$, $a>d$, $v_0\geq \frac b{a-d}u_0\geq 0$, and $h(s)$
is a differentiable nonnegative function on $\mathbb{R}$ has been studied by
Kirane \cite{k1}.

Such equations describe reaction-diffusion processus in physics, chemistry, 
biology and population dynamics. If we have two substances of concentrations 
$u=u(x,t)$ and $v=v(x,t)$ in interaction, the positive
numbers $a$ and $d$ are the so-called diffusion coefficients and 
$b=\vartheta a$, where $\vartheta $ is an arbitrary real number which
describes the drift of the mass of the substance of concentration $v(x,t)$
(cf. \cite{c2, c3,z1}).

Our work is a continuation of the work of Collet and Xin \cite{c1}. Here we
Have a triangular diffusion matrix $(b\neq 0)$. By the same idea we prove
the existence of global solutions to system \eqref{e1.1}.

In the sequel, we use the notation:
\begin{itemize}
\item $\| \cdot \| $ is the supremum norm on $\mathbb{R}^n$:
$\|u\| =\sup_{x\in \mathbb{R}^n}\left| u(x)\right| $

\item For any $\theta \in C_{UB}(\mathbb{R}^n)$, we write
$\int \theta
\equiv \int\nolimits_{\mathbb{R}^n}\theta (x)dx$

\item For $\theta \in L^p(\mathbb{R}^n)$ $(p\geq 1)$, we denote $\|
\theta \|_p^p=\int \left| \theta \right| ^p$.
\end{itemize}

\section{Existence of a Local Solutions and its Positivity}

We convert the system \eqref{e1.1}-\eqref{e1.2} to an abstract first order
system in the
Banach space $X:=C_{UB}(\mathbb{R}^n)\times C_{UB}(\mathbb{R}^n)$ of the 
form
\begin{equation} \label{e2.1}
\begin{gathered}
w^{\prime }(t)=Aw(t)+F(w(t)),\quad t>0, \\
w(0)=w_0\in X\,.
\end{gathered}
\end{equation}
Here
$w(t)=\begin{pmatrix} u(t) \\ v(t)\end{pmatrix}$;
the operator $A$ is defined as
$$
Aw:=\begin{pmatrix} a\Delta & 0 \\  b\Delta & d\Delta \end{pmatrix}
w=( a\Delta u,b\Delta u+d\Delta v),
$$
where $D(A):=\big\{ w=\begin{pmatrix} u \\ v \end{pmatrix}
\in X : \begin{pmatrix} \Delta u \\ \Delta v \end{pmatrix}
\in X\big\}$.
The function $F$ is defined as $F(w(t))=\begin{pmatrix}
-u(t)h(v(t)) \\ u(t)h(v(t))
\end{pmatrix}$.

It is known that for $\lambda >0$ the operator $\lambda \Delta $ generates 
an
analytic semigroup $G(t)$ in the space $C_{UB}(\mathbb{R}^n)$:
\begin{equation} \label{e2.2}
G(t)u=\left( 4\pi \lambda t\right) ^{-n/2}\int_{\mathbb{R}^n}
\exp \big( -\frac{\left| x-y\right| ^2}{4\lambda t}\big) u(y)dy.
\end{equation}
Hence, the operator $A=\begin{pmatrix} a\Delta & 0 \\ b\Delta & d\Delta
\end{pmatrix}$ generates an analytic semigroup defined by
\begin{equation} \label{e2.3}
S(t)=\begin{pmatrix} S_1(t) & 0 \\
\frac b{a-d}\left( S_1(t)-S_2(t)\right) & S_2(t) \end{pmatrix},
\end{equation}
where $S_1(t)$ is the semigroup generated by the operator $a\Delta $, and
$S_2(t)$ is the semigroup generated by the operator $d\Delta $ (See 
\cite{k1}).


Since the map $F$ is locally Lipschitz in $w$ in the space $X$, then proving
the existence of classical solutions on maximal existence interval
$[0,T_0) $ is standard (cf. \cite{h1}, \cite{p1}).

\section{Existence of a Global solution and its Boundedness}

For the existence of a global solution, we use the fact that the
solutions are positive.

\begin{proposition} \label{prop3.1}
Let $(u,v)$ be the solution of the problem \eqref{e1.1}-\eqref{e1.2}
such that
\begin{gather} \label{e3.1}
u_0(x)\geq 0, \quad x\in \mathbb{R}^n,\\
\label{e3.2}
a>d,\quad b>0,\quad v_0(x)\geq \frac b{a-d}u_0(x)\quad \forall x\in
\mathbb{R}^n,
\end{gather}
then
\begin{equation} \label{e3.3}
u(x,t)\geq 0, \quad v(x,t)\geq \frac b{a-d}u(x,t)\quad
\forall ( x,t) \in \mathbb{R}^n\times (0,T_0).
\end{equation}
Moreover, the solution is global and uniformly bounded on 
$\mathbb{R}^n\times
[0,\infty )$. In fact for any $t>0$, we have the estimates:
\begin{gather} \label{e3.4}
\| u(t)\| \leq \| u_0\|\\
\| v(t)\| \leq \left( \frac b{a-d}+\sqrt{\frac ad}
+\frac b{a-d}\sqrt{\frac ad}\right) \| u_0\| +\| v_0\|\,.
\label{e3.5}
\end{gather}
\end{proposition}

\paragraph{Proof.}
The nonnegativity of $u$ is obtained by simple
application of the comparison theorem. Then by the maximum principle we get
\eqref{e3.4}.
To prove $v\geq 0$ under the conditions \eqref{e3.1}-\eqref{e3.2} (see
\cite{k1}).

The solution $(u,v)$ satisfies the integral equations
\begin{gather}\label{e3.6a}
u(x,t)=S_1(t)u_0-\int\nolimits_0^tS_1(t-\tau )u(\tau )v^m(\tau)dt,\\
\label{e3.6b} \begin{aligned}
v(x,t)=&\frac b{a-d}S_1(t)u_0+S_2(t)\big( v_0-\frac b{a-d}u_0\big)
-\frac b{a-d}\int\nolimits_0^tS_1(t-\tau )u(\tau)v^m(\tau )\\
&+\int\nolimits_0^tS_2(t-\tau )\big( u(\tau
)v^m(\tau )+\frac b{a-d}u(\tau )v^m(\tau )\big) d\tau\,.
\end{aligned}
\end{gather}
Here $S_1(t)$ and $S_2(t)$ are the semigroups generated by the operators
$a\Delta $ and $d\Delta $ in the space $C_{UB}(\mathbb{R}^n)$ respectively.
Since $a>d$, using the explicit expression of
$S_1(t-\tau )\left[ u(\tau )v^m(\tau )\right]$
and $S_2(t-\tau )\left[ u(\tau )v^m(\tau )\right]$, one can observe that
\begin{equation}\label{e3.7}
\int_0^tS_2(t-\tau )\left[ u(\tau )v^m(\tau )\right]
d\tau \leq \sqrt{\frac ad}\int\nolimits_0^tS_1(t-\tau )\left[ u(\tau
)v^m(\tau )\right] d\tau .
\end{equation}
On the other hand, using \eqref{e3.6b}, \eqref{e3.7} and the positivity of
the function $u$ given in \eqref{e3.6a}, we deduce the estimate 
\eqref{e3.5}.
Thus, from \eqref{e3.4} and \eqref{e3.5}, we deduce that the solution
$(u,v)$ is global and uniformly bounded on $\mathbb{R}^n\times [0,\infty )$.

In the case where $d>a$, no comparison principle seems to apply.
Nevertheless, we prove the existence of global classical solutions but with
$b<0$.

\begin{theorem} \label{thm3.2}
Let $(u,v)$ be the solution of the problem \eqref{e1.1}-\eqref{e1.2}
satisfying \eqref{e3.1} and
\begin{equation} \label{e3.8}
a<d,\quad b<0,\quad v_0(x)\geq \frac b{a-d}u_0(x) \quad \forall
x\in \mathbb{R}^n,
\end{equation}
then we have \eqref{e3.3} and \eqref{e3.4}.
Moreover,  $(u,v)$ is a global solution.
\end{theorem}

\paragraph{Proof.}
By the same method given in \cite{k1}, we can show that under
the conditions \eqref{e3.1} and \eqref{e3.8} the solution satisfies
\eqref{e3.3} and \eqref{e3.4}. However,
in this case, it is not easy to prove global existence.

To derive estimates of solutions independent of $T_0$, so as to continue the
classical solutions forever in time, we need to some lemmas.

\begin{lemma} \label{lm1}
Let $(u,v)$ be the solution of the system \eqref{e1.1}-\eqref{e1.2}. Define
the functional $L(u,v)=( \alpha +2u-\ln (1+u)) \exp(\varepsilon v)$
with $\alpha ,\varepsilon >0$. Then, for any $\varphi =\varphi (x,t)$ $(x\in
\mathbb{R}^n)$ a smooth nonnegative function with exponential spacial decay 
at
infinity, we have
\begin{multline} \label{e3.9}
\frac d{dt}\int \varphi L\\
\leq \int \left( \varphi_t+d\Delta \varphi \right)
L+\int \varphi (L_2-L_1)uv^m
+ \int \left( (d-a)L_1+bL_2\right) \nabla \varphi \nabla u  \\
-\int \varphi \left[ \left( aL_{11}+bL_{12}\right)
\left| \nabla u\right| ^2+((a+d)L_{12}+bL_{22})\nabla u\nabla
v+dL_{22}\left| \nabla v\right| ^2\right],
\end{multline}
where $L_1=\frac{\partial L}{\partial u}$,
$L_2=\frac{\partial L}{\partial v}$, $L_{11}=\frac{\partial ^2L}{\partial 
u^2}$,
$L_{12}=\frac{\partial ^2L}{\partial u\partial v^{}}$,
$L_{22}=\frac{\partial ^2L}{\partial v^2}$.
\end{lemma}

\paragraph{Proof.}
Note that $L\geq 0$, $L_1\geq 0$, $L_2\geq 0$, $L_{11}\geq 0$,
$L_{12}\geq 0$ and $L_{22}\geq 0$.
We can differentiate under the integral symbol
\begin{equation}\label{e3.10}
\frac d{dt}\int \varphi L=\int \varphi_tL+\int \varphi (L_2-L_1)uv^m
+a\int \varphi L_1\Delta u+b\int \varphi L_1\Delta u+d\int
\varphi L_2\Delta v \,.
\end{equation}
Using integration by parts, we get
\begin{equation} \begin{gathered}
\int \varphi L_1\Delta u=-\int L_1\nabla \varphi \nabla
u-\int \varphi L_{11}\left| \nabla u\right| ^2-\int \varphi L_{12}\nabla
u\nabla v,  \\
\int \varphi L_2\Delta u=-\int L_2\nabla \varphi \nabla
u-\int \varphi L_{12}\left| \nabla u\right| ^2-\int \varphi L_{22}\nabla
u\nabla v, \\
\int \varphi L_2\Delta v=-\int L_2\nabla \varphi \nabla
v-\int \varphi L_{22}\left| \nabla v\right| ^2-\int \varphi L_{12}\nabla
u\nabla v. \end{gathered}\label{e3.11}
\end{equation}
Since $-\int L_2\nabla \varphi \nabla v=\int L\Delta \varphi +\int
L_1\nabla \varphi \nabla u$, using \eqref{e3.11} in \eqref{e3.10} our basic
identity \eqref{e3.9} follows. \hfill$\diamondsuit$

\begin{lemma} In lemma \ref{lm1}, there exist
two positive numbers $\alpha=\alpha \left( a,b,d,\| u_0\| \right) $
and $\varepsilon=\varepsilon \left( a,b,d,\| u_0\| \right) $
depending only on the coefficients $a,b,d$ and the datum $\| u_0\|$,
such that
\begin{equation} \label{e3.12}
\begin{aligned}
\frac d{dt}\int \varphi L\leq &\int \left( \varphi_t+d\Delta \varphi \right)
L-\frac 12\int \varphi L_1uv^m \\
&+\int \left( (d-a)L_1+bL_2\right) \nabla \varphi \nabla u
-\frac 12\int \varphi \left[ \frac a2L_{11}\left| \nabla u\right|
^2+dL_{22}\left| \nabla v\right| ^2\right].
\end{aligned}
\end{equation}
\end{lemma}

\paragraph{Proof.} We choose $\alpha $ and $\varepsilon $ in lemma \ref{lm1}
such that for any
$(u,v) \in \left[ 0,\| u_0\|\right] \times \mathbb{R}^{+}$,
\begin{gather}
L_2\leq \frac 12L_1, \label{e3.13a}\\
(a+d)^2L_{12}^2+b^2L_{22}^2+b(2a+d)L_{12}L_{22}-adL_{11}L_{22}\leq 0
\label{e3.13b}\\
\frac{L_1^2}{L_{11}}\leq L. \label{e3.13c}\\
L_{12}\leq \frac a{2\left| b\right| }L_{11}. \label{e3.13d}
\end{gather}
We verify these conditions as follows. Let
$L_1=(2-\frac 1{1+u})e^{\varepsilon v}$,  and $L_2=\varepsilon \left(
\alpha +2u-\ln (1+u)\right) e^{\varepsilon v}$;
so \eqref{e3.13a} is satisfied if
\begin{equation}
\varepsilon \leq \frac{1+2\| u_0\| }{2(\alpha
+2\| u_0\| )(1+\| u_0\| )}. \label{e3.14}
\end{equation}
We have $L_{11}=e^{\varepsilon v}/(1+u)^2$,
$L_{12}=\varepsilon
(2-\frac 1{1+u})e^{\varepsilon v}$ and $L_{22}=\varepsilon ^2( \alpha
+2u-\ln (1+u)) e^{\varepsilon v}$.

The condition \eqref{e3.13b} is satisfied if
\begin{multline} \label{e3.15}
4(a+d)^2+b^2\varepsilon ^2\left( \alpha +2\| u_0\|
\right) ^2+b(2a+d)\varepsilon \left( \alpha +2\| u_0\| \right)\\
-\frac{ad}{\left( \alpha +2\| u_0\| \right) ^2}
\left( \alpha -\ln (1+\| u_0\| \right) \leq 0.
\end{multline}
This equation is verified if
$b^2\varepsilon ^2\left( \alpha +2\|
u_0\| \right) ^2\leq 1$ and
$$ 4(a+d)^2+\frac{ad}{\left( 1+\|
u_0\| \right) ^2}\left( \alpha -\ln (1+\| u_0\| \right)
+1\leq \alpha \frac{ad}{\left( 1+\| u_0\| \right) ^2}.
$$
Hence we get from these equations that
\begin{gather}
\alpha \geq \ln (1+\| u_0\| )+\frac{(1+\|
u_0\| )^2}{ad}\left( 1+4(a+d)^2\right) , \label{e3.16}\\
\varepsilon \leq \frac 1{\left| b\right| \left( \alpha
+2\| u_0\| \right) }. \label{e3.17}
\end{gather}
Now, to verify \eqref{e3.13c}, It suffices to take
\begin{equation}\label{e3.18}
\alpha \geq \ln (1+\| u_0\| )+\left( 1+2\|
u_0\| \right) ^2.
\end{equation}
To verify \eqref{e3.13d}, it suffices to take
\begin{equation} \label{e3.19}
\varepsilon \leq \frac a{2\left| b\right| \left( 1+\| u_0\|
\right) ^2}.
\end{equation}
Thus, from \eqref{e3.14}, \eqref{e3.16}-\eqref{e3.19}, the real positive
constants $\alpha $ and $\varepsilon $ cited in the lemma are defined by
\begin{gather}
\label{e3.20} \alpha \geq \ln (1+\| u_0\| )+\max \big\{ \frac{%
(1+\| u_0\| )^2}{ad}\left( 1+4(a+d)^2\right) ,\left( 1+2\|
u_0\| \right) ^2.\big\} ,\\
\label{e3.21} \varepsilon \leq \min \big\{ \frac{1+2\| u_0\| }
{2(\alpha +2\| u_0\| )(1+\| u_0\| )},\frac 1{\left|
b\right| \left( \alpha +2\| u_0\| \right) },\frac a{2\left|
b\right| \left( 1+\| u_0\| \right) ^2}\big\}.
\end{gather}
From \eqref{e3.13a} we get
\begin{equation} \label{e3.22}
\int \varphi (L_2-L_1)uv^m\leq -\frac 12\int \varphi L_1uv^m
\end{equation}
and from \eqref{e3.13b}, \eqref{e3.13d} we get
\begin{equation}\label{e3.23}
\begin{aligned}
( aL_{11}&+bL_{12}) \left| \nabla u\right|
^2+((a+d)L_{12}+bL_{22})\nabla u\nabla v+dL_{22}\left| \nabla v\right| ^2\\
&\geq \frac 12\left[ \left( aL_{11}+bL_{12}\right) \left|
\nabla u\right| ^2+dL_{22}\left| \nabla v\right| ^2\right] \\
&\geq \frac 12\left[ \frac a2L_{11}\left| \nabla u\right|
^2+dL_{22}\left| \nabla v\right| ^2\right]. \end{aligned}
\end{equation}
From \eqref{e3.22} and \eqref{e3.23} into \eqref{e3.9} we get our desired
inequality \eqref{e3.12}.
As a consequence of the expressions of $\alpha $, $\varepsilon $ and the
functional $L$, they must be such that $\alpha >16$, $\varepsilon <1/16$.
\hfill$\diamondsuit$

\begin{lemma} \label{lm3}
With the value of $\alpha $ given in \eqref{e3.20} and of $\varepsilon$
given in \eqref{e3.21}, there exist a test function $\varphi $ and two real
positive constants $\beta$ and $\sigma$ such that
\begin{equation} \label{e3.24}
\int \varphi L\leq \beta e^{\sigma t}, \quad \forall t>0.
\end{equation}
\end{lemma}

\paragraph{Proof.} As in \cite{c1}, we define the test function
\begin{equation} \label{e3.25}
\varphi (x)=\frac 1{\left( 1+\left| x-x_0\right| \right) ^2}\,,
\end{equation}
where $x_0$ is  arbitrary  in $\mathbb{R}^n$. It is clear that $\varphi $ is
a smooth function with exponential decay at infinity and satisfies
\begin{equation} \label{e3.26}
\left| \Delta \varphi \right| \leq K\varphi \,,\quad
\left| \nabla\varphi \right| \leq K\varphi \,,
\end{equation}
for some positive constant $K$.

From \eqref{e3.12} with the test function given in \eqref{e3.25} and taking 
in
consideration \eqref{e3.13a}, \eqref{e3.13d}, \eqref{e3.26}, we obtain
\begin{equation}\label{e3.27}
\frac d{dt}\int \varphi L\leq  Kd\int \varphi L+K\left[ (d-a)+\frac 12\left|
b\right| \right] \int L_1\varphi \left| \nabla u\right|
-\frac 14a\int \varphi L_{11}\left| \nabla u\right|^2 .
\end{equation}
We can easily deduce that
\begin{multline} \label{e3.28}
K\big[ (d-a)+\frac 12\left| b\right| \big] \int \varphi
L_1\left| \nabla u\right| -\frac 14a\int \varphi L_{11}\left| \nabla
u\right| ^2 \\
\leq \frac{K^2}a\big( (d-a)+\frac 12\left| b\right| \big)
^2\int \varphi \frac{L_1^2}{L_{11}}.
\end{multline}
Because of \eqref{e3.13c}, $\frac{L_1^2}{L_{11}}\leq L$; hence,
from \eqref{e3.28} and \eqref{e3.27}, we obtain
$$ %\eqref{e3.29}
\frac d{dt}\int \varphi L\leq \big[ Kd+\frac{K^2}a\big( (
d-a) +\frac 12\left| b\right| \big) ^2\big] \int \varphi L.
$$
Thus, we obtain the relation \eqref{e3.24}. More precisely,
$$ \beta \leq \left( \alpha +2\| u_0\| \right)
e^{\varepsilon \| v_0\| }\| \varphi \|_1
\quad\mbox{and}\quad
\sigma =Kd+\frac{K^2}a\big( \left| a-d\right| +\frac 12\left| b\right|
\big) ^2.
$$
\quad\hfill$\diamondsuit$

We use our bound on the nonlinear functional \eqref{e3.24} to control the
$L^p$, $1<p<\infty$, norms of solutions over any unit cube in space.
Here we rely on the fact that the integrand $L$ is exponential in $v$,
and so bounds any powers of $v$ from above.

\begin{lemma} \label{lm4}
For any unit cube $Q$ and any finite $p\geq 1$,
\begin{equation} \label{e3.30}
\int\nolimits_Q\left| v\right| ^pdx\leq 2^n\frac \beta {\alpha
\varepsilon ^p}e^{\sigma t}(p+1)^{p+1}.
\end{equation}
\end{lemma}

\paragraph{Proof.} For any nonnegative integer $p$, we have
\begin{equation} \label{e3.31}
\beta e^{\sigma t}\geq \int \varphi L\geq \alpha \int \varphi
e^{\varepsilon v}\geq \frac{\alpha \varepsilon 
^p}{p!}\int\nolimits_Qv^p\,dx\,.
\end{equation}
By taking $x_0$ as defined in \eqref{e3.25} at the center of $Q$,
we get \eqref{e3.30}. \hfill$\diamondsuit$\smallskip

As consequence of the lemma above, we can prove that there exist two
constants $c=c(n,\lambda )$ and $\omega =\omega (n,\lambda ,\|
u_0\| , \| v_0\|)$ such that
\begin{equation} \label{e3.32}
\begin{aligned}
G(t)u(t)v^m(t)\equiv& \big( 4\pi \lambda t\big) ^{-n/2}
\int_{\mathbb{R}^n}\exp \big( -\frac{\left| x-y\right| ^2}{4\lambda t}\big)
u(y,t)v^m(y,t)dy\\
\leq &c\omega e^{\frac \sigma pt}\varepsilon^{-m}(mp+1)^{m+\frac 1p}
\left( t^{\frac n{2q}}+t^{-\frac n{2p}}\right),
\end{aligned}
\end{equation}
for any $p>\max \left\{ 1,\frac n2\right\} $,  $1/P + 1/q=1$. The inequality
follows. Here $G(t)$ is the semigroup generated by the operator $\lambda
\Delta (\lambda >0)$ on the space $C_{UB}(\mathbb{R}^n)$ (See \cite{c1}).
\hfill$\diamondsuit$

\subsection*{Existence of a Global Solution}
From \eqref{e3.6b} we have
\begin{equation} \label{e3.33}
\begin{aligned}
v(x,t)=&\frac b{a-d}S_1(t)u_0+S_2(t)\big( v_0-\frac b{a-d}u_0\big)
-\frac b{a-d}\int_0^tS_1(t-\tau )u(\tau )v^m(\tau )d\tau \\
& +\big( 1+\frac b{a-d}\big) \int_0^tS_2(t-\tau )u(\tau )v^m(\tau )d\tau .
\end{aligned}
\end{equation}
Since the semigroups $S_1(t)$ and $S_2(t)$ are of contractions and $\frac
b{a-d}>0$, we can deduce
\begin{equation}\label{e3.34a}
\frac b{a-d}S_1(t)u_0+S_2(t)\big( v_0-\frac b{a-d}u_0\big)
\leq \frac b{a-d}\| u_0\| +\| v_0\|.
\end{equation}
Integrating on $\tau \in \left[ 0,t\right] $ and using \eqref{e3.32},
we obtain
\begin{equation}\label{e3.34b}
\begin{aligned}
\int_0^tS_1(t-\tau )u(\tau )v^m(\tau )d\tau
\leq& c_1\omega_1\int\nolimits_0^t\left( (t-\tau )^{\frac n{2q}}
+(t-\tau )^{\frac n{2q}}\right) d\tau \\
\leq& c_1\omega_1\big[ \frac{2q}{2q+n}t^{\frac n{2q}+1}+\frac{%
2p}{2p-n}t^{1-\frac n{2p}}\big],
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.34c}
\int_0^tS_1(t-\tau )u(\tau )v^m(\tau )d\tau \leq c_2\omega
_2\big[ \frac{2q}{2q+n}t^{\frac n{2q}+1}+\frac{2p}{2p-n}t^{1-\frac
n{2p}}\big],
\end{equation}
where $c_1=c(n,a)$, $\omega_1=\omega (n,a,\| u_0\| $,
$\|v_0\| $ $)$, $c_2=c(n,d)$,
$\omega_2=\omega (n,d,\| u_0\| $,
$\| v_0\|)$.
Using \eqref{e3.34a}-\eqref{e3.34c} in \eqref{e3.33}, we get
\begin{equation} \label{e3.35}
\| v(t)\| \leq \frac b{a-d}\| u_0\|
+\| v_0\|  +\big( 1+\frac b{a-d}\big) c_2\omega_2
\big[ \frac{2q}{2q+n}t^{\frac n{2q}+1}+\frac{2p}{2p-n}t^{1-\frac 
n{2p}}\big],
\end{equation}
for all $t\geq 0$, where $p>\frac n2$.
Thus the estimates \eqref{e3.4} and \eqref{e3.35} and the standard parabolic
regularity theory implies the existence of a global classical solution
$(u,v) \in (C([0,+\infty );C_{UB})\cap C^1((0,+\infty );C_{UB}))^2$.
\hfill$\diamondsuit$

\begin{corollary} \label{coro3.3}
Under the assumptions in Theorem \ref{thm3.2},
there exists a classical global solution when the nonlinear reaction term
has the form $uf(v)$, where $f(v)$ is nonnegative continuous in $v\in
\mathbb{R}$
and nondecreasing for $v\geq 0$, $f(0)=\lim_{v\searrow 0^{+}}f(v)=0$,
and $\lim_{v\nearrow \infty }f(v)>0$,
$\lim_{v\nearrow \infty }\frac 1v\log [ f(v)] =0$.
In particular, this form includes the Arrhenius reaction
$uv^m\exp [ -\frac Ev] $, for  $m$ an even nonnegative integer and $E>0$.
\end{corollary}

\paragraph{Proof.}
The estimates in Theorem \ref{thm3.2} remain true for $f(v)$,
since it is bounded from above by the exponential function $e^{\varepsilon
v}$ in the nonlinear functional $L$, thanks to the subexponential growth
condition on $f$. In fact, inequality \eqref{e3.31}, now simply reads
$$
\beta e^{\sigma t}\geq \int \varphi L\geq \alpha \int \varphi
e^{\varepsilon v}\geq \frac \alpha {2^nc_p}\int\nolimits_Q\left[
f(v)\right] ^p dx,
$$
for some constant $c_p$ depending on $p$ and $f$. The remaining estimates
go through as before. Elsewhere we replace $v^m$ by $f(v)$.
\hfill$\diamondsuit$

\section{Remarks}
{\bf (a)} In the system (1.1) we have assumed that $m$ is an even
integer number in order the maximum principle will be applicable, and
consequently the second component $v$ will be positive. The positivity of 
$v$
is used to prove the global existence of solutions. In the other cases of 
$m$
the existence of global solution is unknown.\bigskip\\
{\bf (b)} In the case where $a<d$ and $b>0;$ the possibility of
the existence of a global solution is an open question.



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\end{thebibliography}

\noindent\textsc{Salah BADRAOUI}\\
Universit\'e du 8 Mai 1945-Guelma, \\
Facult\'e des Sciences et Technologie,
Laboratoire LAIG,\\
BP.401, Guelma 24000, Algeria\\
email: s\_badraoui@hotmail.com

\end{document}