\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 61, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/61\hfil Asymptotic behavior of solutions] {Asymptotic behavior of solutions to a\\ $2\times2$ reaction-diffusion system with a cross\\ diffusion matrix on unbounded domains} \author[S. Badraoui\hfil EJDE-2006/61\hfilneg] {Salah Badraoui} \address{Laboratoire LAIG, Universit\'{e} du 08 Mai 1945\\ BP. 401, Guelma 24000, Algeria} \email{sabadraoui@hotmail.com} \date{} \thanks{Submitted October 27, 2005. Published May 11, 2006.} \subjclass[2000]{35B40, 35B45, 35K55, 35K65} \keywords{Reaction-diffusion systems; analytic semi-group; local solution; \hfill\break\indent cross diffusion matrix; unbounded domain; asymptotic behavior of solutions} \begin{abstract} This article concerns the behavior at $\mp \infty$ of solutions to a reaction-diffusion system with a cross diffusion matrix on unbounded domains. We show that the solutions satisfy the free diffusion system for all positive time whenever the initial distribution has limits at $\mp \infty$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} In this paper, we investigate the system of reaction-diffusion equations $$\label{e1} \begin{gathered} u_t=a\frac{\partial ^{2}u}{\partial x^{2}}+\beta \frac{\partial u}{% \partial x}+b\frac{\partial ^{2}v}{\partial x^{2}}+f(t,u,v),\quad x\in \mathbb{R},\; t>0, \\ v_t=c\frac{\partial ^{2}u}{\partial x^{2}}+d\frac{\partial ^{2}v}{\partial x^{2}}+\beta \frac{\partial v}{\partial x}+g(t,u,v),\quad x\in \mathbb{R},\; t>0, \end{gathered}$$ supplemented with the initial conditions $$u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \quad x\in \mathbb{R}.\label{e2}$$ The diffusion coefficients $a$ and $d$ are positive constants while the diffusion coefficients $b,c$ and the coefficient $\beta$ are arbitrary constants. We assume also the following three conditions: \begin{itemize} \item[(H1)] $(a-d)^{2}+4bc>0$, $cd\neq 0$ and $ad>bc$. \item[(H2)] $u_{0},v_{0}\in X$. \item[(H3)] $f(t,u,v)$ and $g(t,u,v)\in X$, for all $t>0$ and $u,v\in X$. Moreover $f$ and $g$ are locally Lipshitz; namely, for all $t_{1}\geq 0$ and all constant $k>0$, there exist a constant $L=L(k,t_{1})>0$ such that $|f(t,w_{1})-f(t,w_{2})|\leq L|w_{1}-w_{2}|,$ is verified for all $w_{1}=(u_{1},v_{1}),$ $w_{2}=( u_{2},v_{2})\in \mathbb{R}\times \mathbb{R}$ with $|w_{1}|\leq k$ , $|w_{2}|\leq k$ and $t\in [0,t_{1}]$. \end{itemize} System \eqref{e1} with specific functional responses has received extensive mathematical treatment since the addition of diffusive terms to the Lotka-Volterra systems. For the case of bounded regions, the questions of existence of globally bounded solutions and their large time behavior have been well studied; various results are presented by Rothe \cite{r1}. Some situations of unbounded regions are presented in \cite{o1}. The system with triangular diffusion matrix $$\label{e*} \begin{gathered} u_t=a\Delta u-uh(v), \quad (x,t)\in \Omega \times (0,\infty ), \\ v_t=b\Delta u+d\Delta v+uh(v), \quad (x,t)\in \Omega \times, (0,\infty ), \end{gathered}$$ 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. In \cite{k1}, He proved that if $a > d > 0$, $b\geq 0$, $b^2<4ad$, the solution $(u,v)$ converges uniformly in $\overline{\Omega}$ to a constant $(k_1,k_2)$ such that $k_1\geq 0$, $k_2\geq0$ and $k_1h(k_2)=0$. Such equations describe reaction-diffusion processus in physics, chemistry, biology and population dynamics. Collet and Xin \cite{c1} have studied the same system \eqref{e*} on $\mathbb{R}^n$ with a diagonal diffusion matrix $(b\neq 0)$ and $h(v)=v^m$, where $m\in \mathbb{N^\star}$. They proved the existence of global solutions and showed that the $L^\infty$ norm of $v$ cannot grow faster than $O(\ln t)$. Also, the system was studied by Avrin \cite{a1} when $b=0$, $v=\exp\{-E/v\}$, $E>0$ and the space variable is in $\mathbb{R}$. The system \eqref{e*} with a triangular diffusion matrix in the case of unbounded domain and $h(v)=v^m$ is studied by Badraoui in \cite{b1,b2}. In \cite{b2} he showed the existence of global classical solution if $v_0(x)\geq \frac b{a-d}u_0(x)$ and $a>d$, $b>0$, or $a<0, b<0$. In \cite{b2} he proved that the $L^\infty$ norm of $v$ cannot grow faster than $O(\ln t)$. Kouachi \cite{k3} obtained a result concerning uniform boundedness of solutions to a system like \eqref{e*} with a general full matrix of diffusion coefficients satisfying a balance law. This result is generalized after by Kouachi \cite{k2} who used the notion of invariant regions and Lyapunov functional. Surprisingly enough, less attention has been given to the behavior of the solutions when the spatial variable $x$ approaches infinity despite the usefulness of this type of result for the numerical treatment of such problems. We are only aware of the article of Gladnov \cite{g1} which generalizes a result of behavior as $x$ approaches infinity of a semi-linear equation posed in $\mathbb{R}^{+}$ studied by Beberns and Fulks \cite{b3}. In this paper, we investigate the behavior of solutions to system \eqref{e1} for large $x$. We show first that the linear operator $A=\begin{pmatrix} a(\cdot )_{xx}+\beta (\cdot )_{x} & b(\cdot )_{xx} \\ c(\cdot )_{xx} & d(\cdot )_{xx}+\beta (\cdot )_{x} \end{pmatrix}$ generates an analytic semi-group over the Banach space $C_{UB}(\mathbb{R})\times C_{UB}(\mathbb{R})$, where $C_{UB}(\mathbb{R})$ is the space of bounded uniformly continuous real-valued functions on $\mathbb{R}$, endowed with the norm of the uniform convergence. After, we show that if the initial conditions $u_{0}$ and $v_{0}$ have finite limits as $x$ approaches $\pm \infty$, the system converges when $x$ approaches $\pm \infty$ to the ordinary differential system associated to it. We will use the following notation: Let $X=(C_{UB}(\mathbb{R}), \| \cdot\|)$ be the space of bounded uniformly continuous real-valued functions on $\mathbb{R}$. For $u: [0,T]\to X$ a continuous function, we use the norm $\|u\|_{1}=\max_{t\in [0,T]}\|u(t)\|.$ For $w=(u,v)\in X\times X$; we define $\|w\|=\|u\|+\|v\|.$ Let $f(t,w)=(f(t,u,v),g(t,u,v))^{t}\equiv \begin{pmatrix} f(t,u,v) \\ g(t,u,v) \end{pmatrix}$. \section{Existence of a local solution} It is well known that for all $\lambda >0$, the linear operator $\lambda \frac{\partial ^{2}}{\partial x^{2}}+\beta \frac{\partial }{\partial x}$ generate analytic semigroup of contractions $G(t)$ on the Banach space. This semigroup is given explicitly by the expression $[ G(t)u] (x)=\frac{1}{\sqrt{4\pi \lambda t}}\int_{\mathbb{R}} \exp (-\frac{|x+\beta t-\xi |^{2}}{4\lambda t})u(\xi )d\xi .$ We recall here that Chen Caisheng \cite{b2} showed that the linear operator $\begin{pmatrix} a\Delta & b\Delta \\ c\Delta & d\Delta \end{pmatrix}$ generates an analytic semigroup of contractions on the space $L^{p}(\Omega )\times L^{p}(\Omega )$ $(1\leq p<\infty )$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$. Inspired by this result, we show that the linear operator $\begin{pmatrix} a(\cdot )_{xx}+\beta (\cdot )_{x} & b(\cdot )_{xx} \\ c(\cdot )_{xx} & d(\cdot )_{xx}+\beta (\cdot )_{x} \end{pmatrix}$ generates an analytic semigroup of contractions on the Banach space $X\times X$. \begin{proposition} \label{prop2.1} Assuming (H1)-(H2), the linear operator $A=\begin{pmatrix} a(\cdot )_{xx}+\beta (\cdot )_{x} & b(\cdot )_{xx} \\ c(\cdot )_{xx} & d(\cdot )_{xx}+\beta (\cdot )_{x} \end{pmatrix}$ generates an analytic semigroup of contractions on the space $X\times X$, given explicitly by $$S(t)= \frac{1}{\lambda _{2}-\lambda _{1}} \begin{pmatrix} (\lambda _{2}-a)S_{1}(t)+(a-\lambda _{1})S_{2}(t) & -bS_{1}(t)+bS_{2}(t) \\ -cS_{1}(t)+cS_{2}(t) & (\lambda _{2}-d)S_{1}(t)+(d-\lambda _{1}) S_{2}(t) \end{pmatrix}, \label{e3}$$ where $\lambda _{1}=\frac{1}{2}(a+d-\sqrt{(a-d)^{2}+4bc}), \quad \lambda _{2}=\frac{1}{2}(a+d+\sqrt{(a-d)^{2}+4bc}),$ and $S_{1}(t)$ and $S_{2}(t)$ are the semigroups generated by the linear operators $\lambda _{1}\frac{\partial ^{2}}{\partial x^{2}}+\beta \frac{\partial }{\partial x}$ and $S_{2}(t)$ respectively. \end{proposition} It should be noted that $\lambda _{1},\lambda _{2}>0$. \begin{proof} It is clear that $S(0)=I$. It is suffices to prove \eqref{e3} for any $w=(u,v)$ in $D(A)=\{ (u,v) : u, v,u_{xx},v_{xx} \in C_{UB}(\mathbb{R})\}.$ We have \begin{itemize} \item[(i)] $\lim_{t\searrow 0}\frac{S(t)w-w}{t}=Aw$, in $X$, \item[(ii)] $S(t+\tau )w=S(t)S(\tau )w$, for any $t,\tau \geq 0$. \end{itemize} In fact, we have \begin{align*} &\lim_{t\searrow 0}\frac{1}{t}\{ S(t)w-w\}\\ &=\frac{1}{\lambda _{2}-\lambda _{1}}\\ &\quad \times \lim_{t\searrow 0} \begin{pmatrix} \frac{1}{t}\{ (\lambda _{2}-a)S_{1}(t)u+(a-\lambda _{1})S_{2}(t)u-u-bS_{1}(t)v+(\lambda _{1}-a)S_{2}(t)v\} \\ \frac{1}{t}\{ -cS_{1}(t)u+cS_{2}(t)u+(\lambda _{2}-d)S_{1}(t)v+( d-\lambda _{1})S_{2}(t)v-v\} \end{pmatrix}. \end{align*} For the first component, we have \begin{align*} &\frac{1}{\lambda _{2}-\lambda _{1}}\lim_{t\searrow 0}\frac{1}{t} \{ (\lambda _{2}-a)S_{1}(t)u+(a-\lambda_{1})S_{2}(t)u-u-bS_{1}(t)v+(\lambda _{1}-a)S_{2}(t)v\}\\ &= \frac{1}{\lambda _{2}-\lambda _{1}}\lim_{t\searrow 0}\{ (\lambda _{2}-a)\frac{S_{1}(t)u-u}{t} +(a-\lambda _{1})\frac{S_{2}(t)u-u}{t}\\ &\quad -b\frac{S_{1}(t)v-v}{t}+b\frac{S_{2}(t)v-v}{t}\} \\ &=\frac{1}{\lambda _{2}-\lambda _{1}}\{ (\lambda _{2}-a)(\lambda _{1}u_{xx}+\beta u_{x})+(a-\lambda _{1})(\lambda _{2}u_{xx}+\beta u_{x})-b(\lambda _{1}v_{xx}+\beta v_{x})\} \\ &\quad +\frac{1}{\lambda _{2}-\lambda _{1}} \{ b(\lambda _{2}v_{xx}+\beta v_{x})\} \\ &= au_{xx}+\beta u_{x}+bv_{xx}, \end{align*} in $C_{UB}(\mathbb{R})$. Similarly, we obtain \begin{align*} &\frac{1}{\lambda _{2}-\lambda _{1}}\lim_{t\searrow 0}\frac{1}{t}% \{ -cS_{1}(t)u+cS_{2}(t)u+(\lambda _{2}-d)S_{1}(t)v+(d-\lambda _{1})S_{2}(t)v-v\} \\ &= cu_{xx}+dv_{xx}+\beta v_{x}, \end{align*} in $C_{UB}(\mathbb{R})$. Therefore (i) is true. Also, by direct computation, we see that (ii) holds. \end{proof} As a consequence of this result we have the following proposition. \begin{proposition} \label{prop2.2} Let (H1)-(H3) be satisfied. Then, the system \eqref{e1}-\eqref{e2} has a unique local solution $(u,v)\in (C[0,T_{0}[,X\times X )$ for some $00$, there is a natural number $n_{0}$ such that for any $m$, $n>n_{0}$ $$\label{e11} \begin{gathered} |u_{0}(y_{m})-u_{0}(y_{n})|<\frac{\varepsilon | \lambda _{2}-\lambda _{1}|}{D}, \\ |u_{0}(z_{m})-v_{0}(z_{n})|<\frac{\varepsilon | \lambda _{2}-\lambda _{1}|}{D}, \\ |v_{0}(y_{m})-v_{0}(y_{n})|<\frac{\varepsilon | \lambda _{2}-\lambda _{1}|}{D}, \\ |v_{0}(z_{m})-v_{0}(z_{n})|<\frac{\varepsilon | \lambda _{2}-\lambda _{1}|}{D}, \end{gathered}$$ where $D=4\max \{ |b|,|c|,|\lambda _{2}-a|,|a-\lambda _{1}|, |\lambda _{2}-d|,|d-\lambda _{1}|\}$. On the other hand, it is easy to show that for any $\varphi \in X$, we have the estimate $$\|\frac{d}{dx}G(t)\varphi \|\leq \frac{\|\varphi \| }{\sqrt{\lambda \pi }}t^{-1/2}, \label{e12}$$ for all $t0$) on $X$, and $\|\Psi \|_{1}=\max_{t\in [0,T]}\|\Psi (t)\|$. Also, from \eqref{e12}, \eqref{e13}, \eqref{e8a}, \eqref{e8b} we get \begin{aligned} &\|\frac{du(t)}{dx}\|\\ &\leq \frac{1}{|\lambda_{2}-\lambda _{1}|}\{ \frac{|\lambda _{2}-a|}{\sqrt{\lambda _{1}\pi }}\|u_{0}\|+\frac{|b|}{\sqrt{\lambda _{1}\pi }}\|v_{0}\|+\frac{|a-\lambda _{1}|}{\sqrt{\lambda _{2}\pi }}\|u_{0}\| +\frac{|b|}{\sqrt{\lambda _{2}\pi }}\|v_{0}\|\} t^{-1/2} \\ &\quad +\frac{2}{|\lambda _{2}-\lambda _{1}|}\{ \frac{|\lambda _{2}-a|}{\sqrt{\lambda _{1}\pi}}\|f\|_{1}+\frac{|b| }{\sqrt{\lambda _{1}\pi }}\|g\|_{1} +\frac{| a-\lambda _{1}|}{\sqrt{\lambda _{2}\pi }}\|f\| _{1}+\frac{|b|}{\sqrt{\lambda _{2}\pi }}\|g\|_{1} \} t^{1/2}, \end{aligned} \label{e14a} % \begin{aligned} &\|\frac{dv(t)}{dx}\|\\ &\leq \frac{1}{|\lambda _{2}-\lambda _{1}|} \{ \frac{|c|}{\sqrt{\lambda _{1}\pi }}\|u_{0}\| +\frac{|\lambda _{2}-d|}{\sqrt{\lambda _{1}\pi }} \|v_{0}\| +\frac{|c|}{\sqrt{\lambda _{2}\pi }}\|u_{0}\|+ \frac{|d-\lambda _{1}|}{\sqrt{\lambda _{2}\pi}}\|v_{0}\| \}t^{-1/2}\\ &\quad +\frac{2}{|\lambda _{2}-\lambda _{1}| }\{\frac{|c|}{\sqrt{\lambda _{1}\pi }}\|f\|_{1} +\frac{|\lambda _{2}-d|}{\sqrt{\lambda _{1}\pi }}\| g\|_{1}+\frac{|c|}{\sqrt{\lambda _{2}\pi }}\|f\|_{1}+ \frac{|d-\lambda _{1}|}{\sqrt{\lambda _{2}\pi}}\|g\|_{1} \}t^{1/2}\,. \end{aligned} \label{e14b} When we set \begin{align*} A=\max \big\{& \frac{1}{|\lambda _{2}-\lambda _{1}|}\{ \frac{| \lambda _{2}-a|}{\sqrt{\lambda _{1}\pi }}\|u_{0}\| +\frac{|b|}{\sqrt{\lambda _{1}\pi }}\|v_{0}\| +\frac{|a-\lambda _{1}|}{\sqrt{\lambda _{2}\pi }}\|u_{0}\| +\frac{|b|}{\sqrt{\lambda _{2}\pi }}\|v_{0}\| \} , \\ &\frac{1}{|\lambda _{2}-\lambda _{1}|}\{ \frac{| c|}{\sqrt{\lambda _{1}\pi }}\|u_{0}\| +\frac{|\lambda _{2}-d|}{\sqrt{\lambda _{1}\pi }}\|v_{0}\| +\frac{|c|}{\sqrt{\lambda _{2}\pi }}\|u_{0}\| +\frac{|d-\lambda _{1}|}{\sqrt{\lambda _{2}\pi }}\|v_{0}\| \} \big\} \end{align*} and \begin{align*} B=\max \big\{& \frac{2}{|\lambda _{2}-\lambda _{1}|}\{ \frac{| \lambda _{2}-a|}{\sqrt{\lambda _{1}\pi }}\|f\|_{1} +\frac{|b|}{\sqrt{\lambda _{1}\pi }}\|g\|_{1} +\frac{|a-\lambda _{1}|}{\sqrt{\lambda _{2}\pi }}\|f\| _{1}+\frac{|b|}{\sqrt{\lambda _{2}\pi }}\|g\| _{1}\} , \\ &\frac{2}{|\lambda _{2}-\lambda _{1}|}\{ \frac{| c|}{\sqrt{\lambda _{1}\pi }}\|f\|_{1}+\frac{| \lambda _{2}-d|}{\sqrt{\lambda _{1}\pi }}\|g\|_{1} +\frac{|c|}{\sqrt{\lambda _{2}\pi }}\|f\|_{1}+\frac{ |d-\lambda _{1}|}{\sqrt{\lambda _{2}\pi }}\|g\|_{1}\} \big\}, \end{align*} we get from \eqref{e14a}-\eqref{e14b}, $$\|\frac{d}{dx}u(t)\|\leq At^{-1/2}+Bt^{1/2},\quad \|\frac{d}{dx}v(t)\|\leq At^{-1/2}+Bt^{1/2}, \label{e15}$$ for all $t\in [0,T]$. \newline Let $k>0$ be a constant such that $\|u\|_{1}\leq k$ and $\|v\|_{1}\leq k$. Using the Lagrange theorem and the estimates \eqref{e15} we obtain $$\label{e16} \begin{gathered} |u(x_{m},t)-u(x_{n},t)| \leq |x_{m}-x_{n}|\| \frac{\partial u}{\partial x}(x',t)\| \leq |x_{m}-x_{n}|\big( At^{-1/2}+Bt^{1/2}\big), \\ |v(x_{m},t)-v(x_{n},t)|\leq |x_{m}-x_{n}| \|\frac{\partial v}{\partial x}(x'',t)\| \leq |x_{m}-x_{n}| \big(At^{-1/2}+Bt^{1/2}\big) \end{gathered}$$ for all $t\in [0,T]$. Here, $x',x''$ are points between $x_{m}$ and $x_{n}$, and $L=L(k,T)>0$ is a constant. On the other hand, we have from (H3) and \eqref{e16}, \begin{align*} &|h_{1}(y_{\tau ,m},\tau )-h_{1}(y_{\tau ,n},\tau )| \\ &\leq |\lambda _{2}-a||f(\tau ,u(y_{\tau ,m},\tau ),v(y_{\tau ,m},\tau ))-f(\tau ,u(y_{\tau ,n},\tau ),v(y_{\tau ,n},\tau ))| \\ &\quad +|b||g(\tau ,u(y_{\tau ,m},\tau ),v(y_{\tau ,m},\tau ))-g(\tau ,u(y_{\tau ,n},\tau ),v(y_{\tau ,n},\tau ))|\\ &\leq L\max \{ |\lambda _{2}-a|,|b|\} \{ |u(y_{\tau ,m},\tau )-u(y_{\tau ,n},\tau )|+| v(y_{\tau ,m},\tau )-v(y_{\tau ,n},\tau )|\} \\ &\leq 2L\max \{ |\lambda _{2}-a|,|b|\} |x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau ^{1/2}), \end{align*} \begin{align*} &|h_{2}(z_{\tau ,m},\tau )-h_{2}(z_{\tau ,n},\tau )|\\ &\leq |a-\lambda _{1}||f(\tau ,u(z_{\tau :m},\tau ),v(z_{\tau :m},\tau ))-f(\tau ,u(z_{\tau :n},\tau ),v(z_{\tau :n},\tau ))|\\ &\quad +|b||g(\tau ,u(z_{\tau :m},\tau ),v(_{\tau ,m},\tau ))-g(\tau ,u(y_{\tau ,n},\tau ),v(y_{\tau ,n},\tau ))| \\ &\leq L\max \{ |a-\lambda _{1}|,|b|\} \{ |u(z_{\tau ,m},\tau )-u(z_{\tau ,n},\tau )|+| v(z_{\tau ,m},\tau )-v(z_{\tau ,n},\tau )|\} \\ &\leq 2L\max \{ |a-\lambda _{1}|,|b|\} |x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau ^{\frac{1}{2}% }), \end{align*} \begin{align*} &|h_{3}(y_{\tau ,m},\tau )-h_{3}(y_{\tau ,n},\tau )| \\ &\leq |c||f(\tau ,u(y_{\tau ,m},\tau ),v(y_{\tau ,m},\tau ))-f(\tau ,u(y_{\tau ,n},\tau ),v(y_{\tau ,n},\tau ))| \\ &\quad +|\lambda _{2}-d||g(\tau ,u(y_{\tau ,m},\tau ),v(y_{\tau ,m},\tau ))-g(\tau ,u(y_{\tau ,n},\tau ),v(y_{\tau ,n},\tau ))|\\ &\leq L\max \{ |c|,|\lambda _{2}-d|\} \{ |u(y_{\tau ,m},\tau )-u(y_{\tau ,n},\tau )| +|v(y_{\tau ,m},\tau )-v(y_{\tau ,n},\tau )|\} \\ &\leq 2L\max \left\{ |c|,|\lambda _{2}-d|\right\} |x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau ^{1/2}), \end{align*} and \begin{align*} &|h_{4}(z_{\tau ,m},\tau )-h_{4}(z_{\tau ,n},\tau )|\\ &\leq |c||f(\tau ,u(z_{\tau ,m},\tau ),v(z_{\tau ,m},\tau ))-f(\tau ,u(z_{\tau ,n},\tau ),v(z_{\tau ,n},\tau ))|\\ &\quad +|d-\lambda _{1}||g(\tau ,u(z_{\tau ,m},\tau ),v(_{\tau ,m},\tau ))-g(\tau ,u(y_{\tau ,n},\tau ),v(y_{\tau ,n},\tau ))|\\ &\leq L\max \{ |c|,|d-\lambda _{1}|\} \{ |u(z_{\tau ,m},\tau )-u(z_{\tau ,n},\tau )| +|v(z_{\tau ,m},\tau )-v(z_{\tau ,n},\tau )|\} \\ &\leq 2L\max \{ |c|,|d-\lambda _{1}|\} |x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau ^{1/2}). \end{align*} % Let $M|\lambda _{2}-\lambda _{1}|=2L\max \left\{ |b| ,|c|,|\lambda _{2}-a|,|a-\lambda _{1}| ,|\lambda _{2}-d|,|d-\lambda _{1}|\right\} \,.$ Then $$\label{e17} \begin{gathered} |h_{1}(y_{\tau ,m},\tau )-h_{1}(y_{\tau ,n},\tau )| \leq M|\lambda _{2}-\lambda _{1}|| x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau^{1/2}), \\ |h_{2}(z_{\tau ,m},\tau )-h_{2}(z_{\tau ,n},\tau )| \leq M|\lambda _{2}-\lambda _{1}|| x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau ^{1/2}), \\ |h_{3}(y_{\tau ,m},\tau )-h_{3}(y_{\tau ,n},\tau )| \leq M|\lambda _{2}-\lambda _{1}|| x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau ^{1/2}) \\ |h_{4}(z_{\tau ,m},\tau )-h_{4}(z_{\tau ,n},\tau )| \leq M|\lambda _{2}-\lambda _{1}|| x_{m}-x_{n}|(A\tau ^{-1/2}+B\tau ^{1/2}). \end{gathered}$$ Inserting \eqref{e11} and \eqref{e17} in \eqref{e10a}-\eqref{e10b}, we get for any $m$, $n>n_{0}$ $$\label{e18} \begin{gathered} |u(x_{m},t)-u(x_{n},t)|\leq \varepsilon +M| x_{m}-x_{n}|(2At^{1/2}+\frac{2}{3}Bt^{3/2}), \\ |v(x_{m},t)-v(x_{n},t)|\leq \varepsilon +M| x_{m}-x_{n}|(2At^{1/2}+\frac{2}{3}Bt^{\frac{3}{2}}), \end{gathered}$$ for all $t\in [0,T]$. Setting \begin{gather*} y_{n}' =y_{\tau ,n}+\beta \tau +2\eta \sqrt{\lambda _{1}\tau }, \quad y_{\sigma ,n}'=y_{\tau ,n}+\beta (\tau -\sigma )+2\eta \sqrt{\lambda _{1}(\tau -\sigma )}, \\ z_{n}' =z_{\tau ,n}+\beta \tau +2\eta \sqrt{\lambda _{2}\tau }, \quad z_{\sigma ,n}'=z_{n,\tau }+\beta (\tau -\sigma )+2\eta \sqrt{\lambda _{2}(\tau -\sigma )}. \end{gather*} Then, from \eqref{e11} and \eqref{e17} into \eqref{e10a}-\eqref{e10b}, we obtain \begin{align*} &|\lambda _{2}-\lambda _{1}||u(y_{\tau ,m},\tau )-u(y_{\tau ,n},\tau )|\\ &\leq \frac{|\lambda _{2}-a|}{\sqrt{\pi }} \int_{\mathbb{R}}e^{-\eta ^{2}}|u_{0}(y_{m}')-u_{0}(y_{n}')|d\eta +\frac{|b|}{\sqrt{\pi}}\int_{\mathbb{R}}e^{-\eta ^{2}}|v_{0}(y_{m}')-v_{0}(y_{n}')|d\eta \\ &\quad +\frac{|a-\lambda _{1}|}{\sqrt{\pi }}\int_{\mathbb{R} }e^{-\eta ^{2}}|u_{0}(z_{m}')-u_{0}(z_{n}')|d\eta +\frac{|b|}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}}|v_{0}(z_{m}')-v_{0}(z_{n}')|d\eta \\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{\tau }\int_{\mathbb{R} }e^{-\eta ^{2}}|h_{1}(y_{\sigma ,m}',\sigma )-h_{1}(y_{\sigma ,n}',\sigma )|d\eta d\sigma\\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{\tau }\int_{\mathbb{R} }e^{-\eta ^{2}}|h_{2}(z_{\sigma ,m}',\sigma )-h_{2}(z_{\sigma ,n}',\sigma )|d\eta d\sigma \\ &\leq \varepsilon |\lambda _{2}-\lambda _{1}|+M|\lambda _{2}-\lambda _{1}||x_{m}-x_{n}|(2A\tau ^{\frac{1}{2}} +\frac{2}{3}B\tau ^{\frac{3}{2}}) \end{align*} and \begin{align*} &|\lambda _{2}-\lambda _{1}||v(z_{\tau ,m},\tau )-v(z_{\tau ,n},\tau )|\\ &\leq \frac{|c|}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}}|u_{0}(y_{m}')-u_{0}(y_{n}')|d\eta + \frac{|\lambda _{2}-d|}{\sqrt{\pi }}\int_{\mathbb{R}% }e^{-\eta ^{2}}|v_{0}(y_{m}')-v_{0}(y_{\sigma ,n}')|d\eta \\ &\quad +\frac{|c|}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}}|u_{0}(z_{m}')-u_{0}(z_{n}')|d\eta +\frac{|d-\lambda _{1}|}{\sqrt{\pi }}\int_{\mathbb{R}% }e^{-\eta ^{2}}|v_{0}(z_{m}')-v_{0}(z_{n}')| d\eta \\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{\tau }\int_{\mathbb{R}% }e^{-\eta ^{2}}|h_{3}(y_{\sigma ,m}',\sigma )-h_{3}(y_{\sigma ,n}',\sigma )|d\eta d\sigma \\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{t\tau }\int_{\mathbb{R}% }e^{-\eta ^{2}}|h_{4}(z_{\sigma ,m}',\sigma )-h_{4}(z_{\sigma ,n}',\sigma )|d\eta d\sigma \\ &\leq \varepsilon |\lambda _{2}-\lambda _{1}|+M|\lambda _{2}-\lambda _{1}||x_{m}-x_{n}|(2A\tau ^{\frac{1}{2% }}+\frac{2}{3}B\tau ^{\frac{3}{2}}). \end{align*} Whence \begin{gather} |u(y_{\tau ,m},\tau )-u(y_{\tau ,n},\tau )|\leq \varepsilon +M|x_{m}-x_{n}|(2A\tau ^{1/2}+\frac{2}{3}B\tau ^{% \frac{3}{2}})\label{e19a} \\ |v(z_{\tau ,m},\tau )-v(z_{\tau ,n},\tau )|\leq \varepsilon +M|x_{m}-x_{n}|(2A\tau ^{1/2}+\frac{2}{3}B\tau ^{% \frac{3}{2}}), \label{e19b} \end{gather} and from \eqref{e19a}-\eqref{e19b} in \eqref{e10a}-\eqref{e10b} we get $$\label{e20} \begin{gathered} |u(y_{m},t)-u(y_{n},t)|\leq \varepsilon (1+Mt)+M^{2}| x_{m}-x_{n}|(\frac{2^{2}}{3}At^{\frac{3}{2}} +\frac{2^{2}}{3\times 5}Bt^{\frac{5}{2}}) \\ |v(z_{m},t)-u(z_{n},t)|\leq \varepsilon (1+Mt)+M^{2}| x_{m}-x_{n}|(\frac{2^{2}}{3}At^{\frac{3}{2}} +\frac{2^{2}}{3\times 5}Bt^{\frac{5}{2}}), \end{gathered}$$ for all $t\in [0,T]$. Iterating this operation $N$ times we obtain \begin{align*} |u(x_{m},t)-u(x_{n},t)| &\leq \varepsilon \big(1+Mt+\frac{(Mt)^{2}}{2!}\dots \frac{(Mt)^{n-1}}{(N-1)!}\big)\\ &\quad +|x_{m}-x_{n}|\Big( \frac{(2M)^{N}}{1\times 3\times 5\times \dots \times (2N-1)}At^{N-\frac{1}{2}}\\ &\quad +\frac{(2M)^{N}}{1\times 3\times 5\times \dots \times (2N+1)}Bt^{N+\frac{1}{2}}\Big), \end{align*} and \begin{align*} |v(x_{m},t)-v(x_{n},t)| &\leq \varepsilon \big(1+Mt+\frac{(Mt)^{2}}{2!}\dots \frac{(Mt)^{n-1}}{(N-1)!}\big)\\ &\quad +|x_{m}-x_{n}|\big( \frac{(2M)^{N}}{1\times 3\times 5\times \dots \times (2N-1)}\\ &\quad\times At^{N-\frac{1}{2}}\frac{(2M)^{N}}{1\times 3\times 5\times \dots \times (2N+1)}Bt^{N+\frac{1}{2}} \big). \end{align*} Passing to the limit when $N$ approaches infinity, we obtain $$\label{e21} |u(x_{m},t)-u(x_{n},t)|\leq \varepsilon e^{Mt}, \quad |v(x_{m},t)-v(x_{n},t)|\leq \varepsilon e^{Mt},$$ for all $t\in [0,T]$. From these inequalities, we deduce that the sequences $(u(x_{n},t))_{n}$ and $(v(x_{n},t))_{n}$ are Cauchy sequences of continuous functions from $[ 0,T]$ into $X$, hence they converge uniformly on $[ 0,T]$ to some continuous functions $U$ and $V$, respectively. The solution $(u,v)$ satisfies the system of integral equation \begin{align*} &(\lambda _{2}-\lambda _{1})u(x,t) \\ &=\frac{1}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}} [ (\lambda _{2}-a)u_{0}-bv_{0}] (y,t)d\eta +\frac{1}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}} [(a-\lambda _{1})u_{0}+bv_{0}] (z,t)d\eta \\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{1}(y_{\tau },\tau )d\eta d\tau +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{2}(z_{\tau },\tau )d\eta d\tau , \end{align*} % \begin{align*} &(\lambda _{2}-\lambda _{1})v(x,t) \\ &= \frac{1}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}} [ -cu_{0}+(\lambda _{2}-d)v_{0}] (y,t)d\eta +\frac{1}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}} [cu_{0}+(d-\lambda _{1})v_{0}] (z,t)d\eta \\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{3}(y_{\tau },\tau )d\eta d\tau +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{4}(z_{\tau },\tau )d\eta d\tau . \end{align*} With the previous substitution of the spatial variable, and for any sequence $(x_{n})_{n}$ tending to $+\infty$, we have \begin{aligned} &(\lambda _{2}-\lambda _{1})u(x_{n},t) \\ &=\frac{1}{\sqrt{\pi }}\int_{\mathbb{R}} e^{-\eta ^{2}}[ (\lambda_{2}-a)u_{0}-bv_{0}] (y_{n},t)d\eta +\frac{1}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}}[ (a-\lambda _{1})u_{0}+bv_{0}] (z_{n},t)d\eta \\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{1}(y_{\tau ,n},\tau )d\eta d\tau +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{2}(z_{\tau ,n},\tau )d\eta d\tau, \end{aligned} \label{e22a} \begin{aligned} &(\lambda _{2}-\lambda _{1})v(x_{n},t) \\ &=\frac{1}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}} [ -cu_{0}+(\lambda _{2}-d)v_{0}] (y_{n},t)d\eta\\ &\quad +\frac{1}{\sqrt{\pi }}\int_{\mathbb{R}}e^{-\eta ^{2}} [cu_{0}+(d-\lambda _{1})v_{0}] (z_{n},t)d\eta \\ &\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{3}(y_{\tau ,n},\tau )d\eta d\tau +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R} }e^{-\eta ^{2}}h_{4}(z_{\tau ,n},\tau )d\eta d\tau . \end{aligned}\label{e22b} By the dominated convergence theorem we have $$\label{e23} \begin{gathered} \lim_{n\to \infty }\int_{\mathbb{R}}e^{-\eta ^{2}}[ (\lambda _{2}-a)u_{0}-bv_{0}] (y_{n},t)d\eta =\sqrt{\pi } \{ (\lambda _{2}-a)U_{0}-bV_{0}\} , \\ \lim_{n\to \infty }\int_{\mathbb{R}}e^{-\eta ^{2}}[ (a-\lambda _{1})u_{0}+bv_{0}] (z_{n},t)d\eta =\sqrt{\pi } \{ (a-\lambda _{1})U_{0}+bV_{0}\} , \\ \lim_{n\to \infty }\int_{\mathbb{R}}e^{-\eta ^{2}}[ -cu_{0}+(\lambda _{2}-d)v_{0}] (y_{n},t)d\eta =\sqrt{\pi } \{ -cU_{0}+(\lambda _{2}-d)V_{0}\} , \\ \lim_{n\to \infty }\int_{\mathbb{R}}e^{-\eta ^{2}}[ cu_{0}+(d-\lambda _{1})v_{0}] (z_{n},t)d\eta =\sqrt{\pi } \{ cU_{0}+(d-\lambda _{1})V_{0}\} , \end{gathered}$$ where $U_{0}=\lim_{n\to \infty }u_{0}(x_{n})$ and $V_{0}=\lim_{n\to \infty }v_{0}(x_{n})$. We also have $|e^{-\eta ^{2}}h_{i}(y_{\tau ,n},\tau )|\leq C(T)e^{-\eta ^{2}},$ for $i=1,2,3,4$ and all $0\leq \tau \leq t\leq T$, where $C(T)=\max \big\{ |\lambda _{2}-a|,|b|,| a-\lambda _{1}|,|c|,|d-\lambda _{1}| \big\} (\|f\|_{1}+\|g\|_{1})$ Using again the dominated convergence theorem, we obtain \begin{aligned} &\lim_{n\to \infty }\int_{0}^{t}\int_{\mathbb{R}}e^{-\eta ^{2}}h_{1}(y_{\tau ,n},\tau )\\ &= \sqrt{\pi }\int_{0}^{t}\{ (\lambda _{2}-a)f( \tau ,U(\tau ),v(\tau ))-bg(\tau ,U(\tau ),v(\tau))\} d\tau, \end{aligned}\label{e24a} \begin{aligned} &\lim_{n\to \infty }\int_{0}^{t}\int_{% \mathbb{R}}e^{-\eta ^{2}}h_{2}(y_{\tau ,n},\tau ) \\ &=\sqrt{\pi }\int_{0}^{t}\{ (a-\lambda _{1})f( \tau ,U(\tau ),v(\tau ))+bg(\tau ,U(\tau ),v(\tau))\} d\tau. \end{aligned}\label{e24b} We have also \begin{gather} \begin{aligned} &\lim_{n\to \infty }\int_{0}^{t}\int_{\mathbb{R}} e^{-\eta ^{2}}h_{3}(y_{\tau ,n},\tau ) \\ &=\sqrt{\pi }\int_{0}^{t}\{ -cf(\tau ,U(\tau ),v(\tau )) +(\lambda _{2}-d)g(\tau ,U(\tau ),v(\tau))\} d\tau, \end{aligned} \label{e24c}\\ \begin{aligned} &\lim_{n\to \infty }\int_{0}^{t}\int_{\mathbb{R}} e^{-\eta ^{2}}h_{4}(y_{\tau ,n},\tau ) \\ &=\sqrt{\pi }\int_{0}^{t}\{ cf(\tau ,U(\tau),v(\tau )) +(d-\lambda _{1})g(\tau ,U(\tau ),v(\tau ))\} d\tau. \end{aligned}\label{e24d} \end{gather} Thanks to \eqref{e23} and \eqref{e24a}-\eqref{e24d}, if we pass to the limit in \eqref{e22a}-\eqref{e22b}, we obtain \begin{gather*} U(t)=U_{0}+\int_{0}^{t}f(\tau ,U(\tau ),V(\tau ))d\tau , \\ V(t)=V_{0}+\int_{0}^{t}g(\tau ,U(\tau ),V(\tau ))d\tau , \end{gather*} for all $0\leq t\leq T$. The ordinary differential system then follows. \end{proof} We remark remark that the same analysis holds for $u_{0},v_{0}\in C_{-}\equiv \{ u\in X: \lim_{x\to -\infty }u(x)\text{ exist}\} .$ \subsection*{Conclusions} We have proved the result of asymptotic behavior when $x\to \infty$ thanks to the explicit expression of the semigroup generated by the linear operator $A=\begin{pmatrix} a(.)_{xx}+\beta (.)_{x} & b(.)_{xx} \\ c(.)_{xx} & d(.)_{xx}+\lambda (.)_{x} \end{pmatrix},$ where $\lambda =\beta$ in the space $X^{2}$, where $X=(C_{UB}(\mathbb{R)},\|.\|)$ under some conditions over the coefficients $a,b,c$ and $d$. The analytic expression of the semigroup generated by the operator $A$ if $\lambda \neq \beta$ still an open problem. \begin{thebibliography}{9} \bibitem{a1} J. D. Avrin; \emph{Qualitative Theory for a Model of Laminar Flames with Arbitrary Nonnegative Initial Data}, J. 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