\documentclass[twoside]{article} \usepackage{amssymb,amsmath} \pagestyle{myheadings} \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 $a0$, $d>0$, $b\neq 0$, $m$ 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 $$\label{e2.1} \begin{gathered} w^{\prime }(t)=Aw(t)+F(w(t)),\quad t>0, \\ w(0)=w_0\in X\,. \end{gathered}$$ 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 $C_{UB}(\mathbb{R}^n)$: $$\label{e2.2} G(t)u=\left( 4\pi \lambda t\right) ^{-n/2}\int_{\mathbb{R}^n} \exp \big( -\frac{\left| x-y\right| ^2}{4\lambda t}\big) u(y)dy.$$ Hence, the operator $A=\begin{pmatrix} a\Delta & 0 \\ b\Delta & d\Delta \end{pmatrix}$ generates an analytic semigroup defined by $$\label{e2.3} S(t)=\begin{pmatrix} S_1(t) & 0 \\ \frac b{a-d}\left( S_1(t)-S_2(t)\right) & S_2(t) \end{pmatrix},$$ 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 $$\label{e3.3} u(x,t)\geq 0, \quad v(x,t)\geq \frac b{a-d}u(x,t)\quad \forall ( x,t) \in \mathbb{R}^n\times (0,T_0).$$ 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 $$\label{e3.7} \int_0^tS_2(t-\tau )\left[ u(\tau )v^m(\tau )\right] d\tau \leq \sqrt{\frac ad}\int\nolimits_0^tS_1(t-\tau )\left[ u(\tau )v^m(\tau )\right] d\tau .$$ 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 \label{e3.8} a0$. Then, for any$\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 $$\label{e3.10} \frac d{dt}\int \varphi L=\int \varphi_tL+\int \varphi (L_2-L_1)uv^m +a\int \varphi L_1\Delta u+b\int \varphi L_1\Delta u+d\int \varphi L_2\Delta v \,.$$ Using integration by parts, we get $$\begin{gathered} \int \varphi L_1\Delta u=-\int L_1\nabla \varphi \nabla u-\int \varphi L_{11}\left| \nabla u\right| ^2-\int \varphi L_{12}\nabla u\nabla v, \\ \int \varphi L_2\Delta u=-\int L_2\nabla \varphi \nabla u-\int \varphi L_{12}\left| \nabla u\right| ^2-\int \varphi L_{22}\nabla u\nabla v, \\ \int \varphi L_2\Delta v=-\int L_2\nabla \varphi \nabla v-\int \varphi L_{22}\left| \nabla v\right| ^2-\int \varphi L_{12}\nabla u\nabla v. \end{gathered}\label{e3.11}$$ 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 \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{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 $$\varepsilon \leq \frac{1+2\| u_0\| }{2(\alpha +2\| u_0\| )(1+\| u_0\| )}. \label{e3.14}$$ 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 $$\label{e3.18} \alpha \geq \ln (1+\| u_0\| )+\left( 1+2\| u_0\| \right) ^2.$$ To verify \eqref{e3.13d}, it suffices to take $$\label{e3.19} \varepsilon \leq \frac a{2\left| b\right| \left( 1+\| u_0\| \right) ^2}.$$ 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 $$\label{e3.22} \int \varphi (L_2-L_1)uv^m\leq -\frac 12\int \varphi L_1uv^m$$ and from \eqref{e3.13b}, \eqref{e3.13d} we get \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} 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 $$\label{e3.24} \int \varphi L\leq \beta e^{\sigma t}, \quad \forall t>0.$$ \end{lemma} \paragraph{Proof.} As in \cite{c1}, we define the test function $$\label{e3.25} \varphi (x)=\frac 1{\left( 1+\left| x-x_0\right| \right) ^2}\,,$$ 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 $$\label{e3.26} \left| \Delta \varphi \right| \leq K\varphi \,,\quad \left| \nabla\varphi \right| \leq K\varphi \,,$$ 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 $$\label{e3.27} \frac d{dt}\int \varphi L\leq Kd\int \varphi L+K\left[ (d-a)+\frac 12\left| b\right| \right] \int L_1\varphi \left| \nabla u\right| -\frac 14a\int \varphi L_{11}\left| \nabla u\right|^2 .$$ 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\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$\diamondsuit\subsection*{Existence of a Global Solution} From \eqref{e3.6b} we have \label{e3.33} \begin{aligned} v(x,t)=&\frac b{a-d}S_1(t)u_0+S_2(t)\big( v_0-\frac b{a-d}u_0\big) -\frac b{a-d}\int_0^tS_1(t-\tau )u(\tau )v^m(\tau )d\tau \\ & +\big( 1+\frac b{a-d}\big) \int_0^tS_2(t-\tau )u(\tau )v^m(\tau )d\tau . \end{aligned} Since the semigroupsS_1(t)$and$S_2(t)$are of contractions and$\frac b{a-d}>0$, we can deduce $$\label{e3.34a} \frac b{a-d}S_1(t)u_0+S_2(t)\big( v_0-\frac b{a-d}u_0\big) \leq \frac b{a-d}\| u_0\| +\| v_0\|.$$ Integrating on$\tau \in \left[ 0,t\right] and using \eqref{e3.32}, we obtain \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} and $$\label{e3.34c} \int_0^tS_1(t-\tau )u(\tau )v^m(\tau )d\tau \leq c_2\omega _2\big[ \frac{2q}{2q+n}t^{\frac n{2q}+1}+\frac{2p}{2p-n}t^{1-\frac n{2p}}\big],$$ wherec_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 $$\label{e3.35} \| v(t)\| \leq \frac b{a-d}\| u_0\| +\| v_0\| +\big( 1+\frac b{a-d}\big) c_2\omega_2 \big[ \frac{2q}{2q+n}t^{\frac n{2q}+1}+\frac{2p}{2p-n}t^{1-\frac n{2p}}\big],$$ 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$a0;$the possibility of the existence of a global solution is an open question. \begin{thebibliography}{9} \frenchspacing \bibitem{a1} J. D. Avrin, Qualitative theory for a model of laminar flames with arbitrary nonnegative initial data, J.D.E, 84(1990), 209-308. \bibitem{c1} P. Collet \& J. Xin, Global Existence and Large Time Asymptotic Bounds of$L^\infty $Solutions of Thermal Diffusive Combustion Systems on$\mathbb{R}^n\$, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 23(1996), 625-642 . \bibitem{c2} E. L. Cussler, Multicomponent Diffusion, Elsevier Publishing Company, Amsterdam, 1976. \bibitem{c3} E. L. Cussler, Diffusion, Cambridge University Press, Second Edition, 1997. \bibitem{h1} D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, New York, 1981. \bibitem{k1} M. Kirane, Global Bounds and Asymptotics for a System of Reaction-Diffusion Equations, Journal of Mathematical Analysis and Applications 138, pp. 328-342, 1989. \bibitem{l1} B. Larrouturou, The equations of one-dimentional unsteady flames propagation: existence and uniqueness, SIAM J. Math. Anal. 19, pp. 32-59, 1988. \bibitem{p1} A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. \bibitem{z1} E. Zeidler, Nonlinear Functional Analysis and its Applications, Tome II/B, Springer Verlag, 1990. \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} 