\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
\begin{equation} \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}
\end{equation}
supplemented with the initial conditions
\begin{equation}
u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \quad x\in \mathbb{R}.\label{e2}
\end{equation}
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
\begin{equation} \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}
\end{equation}
 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
\begin{equation}
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}
\end{equation}
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 $0<T_{0}$ $<\infty $.
\end{proposition}

 \begin{proof}  It suffices to set
\begin{gather*}
A=\begin{pmatrix}
a(\cdot )_{xx}+\beta (\cdot )_{x} & b(\cdot )_{xx} \\
c(\cdot )_{xx} & d(\cdot )_{xx}+\beta (\cdot )_{x}
\end{pmatrix},
\\
w_{0}=(u_{0},v_{0})^{t}.
\end{gather*}
Then, the system \eqref{e1}, \eqref{e2}  is written as
\begin{gather}
w_{t}=Aw+F(t,w), \label{e4} \\
w(0)=w_{0}.  \label{e5}
\end{gather}
Taking into account  \cite[proposition 5.1, theorem 6.1.4]{p1},
the proof is complete.
\end{proof}
 Let
\[
C_{\pm }:=\{ u\in X: \lim_{x\to \pm
\infty }u(x)\text{ exist}\} .
\]

\section{Behavior of  solutions as $x \to \infty $}

 It turns out that if $u_{0},v_{0}\in C_{\pm }$ then the
diffusive system, for $x$ large, will behave like the system of
ordinary differential equations associated to it, and hence, for
$x$ large, it can be replaced by the latter which is  simpler
to analyze.

 For instance, for the numerical treatment of system
\eqref{e1}-\eqref{e2}, one
can develop a numerical scheme for an approximated problem through a
truncated domain $[-R,R] $ and use the system of ordinary
differential equations in $\mathbb{R}\backslash [ -R,R] $.

\begin{theorem} \label{thm3.1}
Under the assumptions (H1)-(H3),
if $u_{0},v_{0}\in C_{+}$ , then $u(t),v(t)\in C_{+}$, for all
 $t\in [0,t[$ where $t<t_{\rm max}$. Moreover,
$U(t)\equiv \lim_{x\to +\infty }u(x,t)$ and
$V(t)\equiv \lim_{x\to +\infty }v(x,t)$ satisfy the  system of
ordinary differential equations
\begin{equation} \label{e6}
\begin{gathered}
U'(t)=f(t,U(t),V(t)),\\
V'(t)=g(t,U(t),V(t)),
\end{gathered}
\end{equation}
for any $t<t_{\rm max}$, with the initial data
\begin{equation}
U(0)=\lim_{x\to +\infty }u_{0}(x),\quad
V(0)=\lim_{x\to +\infty }v_{0}(x).  \label{e7}
\end{equation}
\end{theorem}

\begin{proof} The solution $(u,v)$
satisfies the system of integral forms
\begin{equation}
\begin{aligned}
(\lambda _{2}-\lambda _{1})u(t)
&=S_{1}(t)((\lambda_{2}-a)u_{0}-bv_{0})+S_{2}(t)((a-\lambda
_{1})u_{0}+bv_{0})
 \\
&\quad +\int_{0}^{t}S_{1}(t-\tau )((\lambda
_{2}-a)f(\tau ,u,v)-bg(\tau ,u,v))d\tau
\\
&\quad +\int_{0}^{t}S_{2}(t-\tau )((a-\lambda
_{1})f(\tau ,u,v)+bg(\tau ,u,v))d\tau ,
\end{aligned} \label{e8a}
\end{equation}
%
\begin{equation}
\begin{aligned}
 (\lambda _{2}-\lambda _{1})v(t)
&=S_{1}(t)(-cu_{0}+(\lambda
_{2}-d)v_{0})+S_{2}(t)(cu_{0}+(d-\lambda_{1})v_{0})
\\
&\quad +\int_{0}^{t}S_{1}(t-\tau )(-cf(\tau,u,v)+(\lambda _{2}-d)
g(\tau ,u,v))d\tau\\
&\quad +\int_{0}^{t}S_{2}(t-\tau )(cf(\tau ,u,v)+(d-\lambda _{1})
g(\tau ,u,v))d\tau .
\end{aligned}\label{e8b}
\end{equation}
Changing the spatial variable, $u$ and $v$ can be written as
\begin{equation}
\begin{aligned}
&(\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{aligned}
\label{e9a}
\end{equation}
%
\begin{equation}
\begin{aligned}
&(\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{aligned}
\label{e9b}
\end{equation}
where
\begin{gather*}
y = x+\beta t+2\eta \sqrt{\lambda _{1}t}, \\
z = x+\beta t+2\eta \sqrt{\lambda _{2}t}, \\
y_{\tau } = x+\beta (t-\tau )+2\eta \sqrt{\lambda _{1}t}, \\
z_{\tau } = x+\beta (t-\tau )+2\eta \sqrt{\lambda _{2}(t-\tau )},
\end{gather*}
 and
\begin{gather*}
h_{1}(y_{\tau },\tau ) =((\lambda _{2}-a)f(.,u,v)-bg(.,u,v))
(y_{\tau },\tau ), \\
h_{2}(z_{\tau },\tau ) =((a-\lambda _{1})f(.,u,v)+bg(.,u,v))
(z_{\tau },\tau ), \\
h_{3}(y_{\tau },\tau ) = (-cf(.,u,v)+(\lambda _{2}-d)g(.,u,v))
(y_{\tau },\tau ),\\
h_{4}(z_{\tau },\tau ) = (cf(.,u,v)+(d-\lambda _{1})g(.,u,v))
(z_{\tau },\tau ).
\end{gather*}

 To show that $u$ and $v$ have limits when $x\to +\infty $,
for any positive $t<t_{\rm max}$, it suffices to verify that for
any sequence of real numbers $(x_{n})_{n}$ satisfying
$\lim_{n\to \infty }x_{n}=+\infty $, the sequences
$(u(x_{n},t))_{n\geq 1}$ and $(v(x_{n},t))_{n\geq 1}$ are
Cauchy sequences in $\mathbb{R}$.
To do so, let $t<t_{\rm max}$, and set
\begin{gather*}
y_{n} =x_{n}+\beta t+2\eta \sqrt{\lambda _{1}t}, \quad
y_{\tau ,n}=x_{n}+\beta (t-\tau )+2\eta \sqrt{\lambda _{1}(t-\tau )}, \\
z_{n} =x_{n}+\beta t+2\eta \sqrt{\lambda _{2}t},\quad
z_{\tau ,n}=x_{n}+\beta (t-\tau )+2\eta \sqrt{\lambda _{2}(t-\tau )}.
\end{gather*}
Then from \eqref{e9a}--\eqref{e9b}, we get
\begin{equation}
\begin{aligned}
&|\lambda _{2}-\lambda _{1}||u(x_{m},t)-u(x_{n},t)| \\
&\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}^{t}\int_{\mathbb{R}
}e^{-\eta ^{2}}|h_{3}(y_{\tau ,m},\tau )-h_{3}(y_{\tau ,n},\tau
)|d\eta d\tau\\
&\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R}
}e^{-\eta ^{2}}|h_{4}(z_{\tau ,m},\tau )-h_{4}(z_{\tau
,n},\tau )|d\eta d\tau.
\end{aligned}  \label{e10a}
\end{equation}
%
\begin{equation}
\begin{aligned}
&|\lambda _{2}-\lambda _{1}||v(x_{m},t)-v(x_{n},t)| \\
&\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_{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}^{t}\int_{\mathbb{R}
}e^{-\eta ^{2}}|h_{3}(y_{\tau ,m},\tau )-h_{3}(y_{\tau ,n},\tau
)|d\eta d\tau \\
&\quad +\frac{1}{\sqrt{\pi }}\int_{0}^{t}\int_{\mathbb{R}
}e^{-\eta ^{2}}|h_{4}(z_{\tau ,m},\tau )-h_{4}(z_{\tau
,n},\tau )|d\eta d\tau.
\end{aligned}  \label{e10b}
\end{equation}
 Since $u_{0},v_{0}\in C_{+}$, for any positive $\varepsilon >0$,
there is a natural number $n_{0}$ such that for any $m$, $n>n_{0}$
\begin{equation} \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}
\end{equation}
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
\begin{equation}
\|\frac{d}{dx}G(t)\varphi \|\leq \frac{\|\varphi \|
}{\sqrt{\lambda \pi }}t^{-1/2},  \label{e12}
\end{equation}
for all $t<t_{\rm max}$ (see Appendix). Hence, for all
continuous function $\Psi :[0,T]\to X$, we have
\begin{equation}
\|\frac{d}{dx}\int_{0}^{t}G(t-\tau )\Psi (\tau )d\tau
\|\leq 2\frac{\|\Psi \|_{1}}{\sqrt{\lambda \pi }}t^{-1/2},\newline
\label{e13}
\end{equation}
for all $t\in [0,T]$, where $T<t_{\rm max}$.

 Here, $G(t)$ is the semigroup generated by the operator
$\lambda\Delta $ ($\lambda >0$) 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{equation}
\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}
\end{equation}
%
\begin{equation}
\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}
\end{equation}
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},
\begin{equation}
\|\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}
\end{equation}
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
\begin{equation} \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}
\end{equation}
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
\begin{equation} \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}
\end{equation}
Inserting \eqref{e11} and \eqref{e17} in \eqref{e10a}-\eqref{e10b}, we get
for any $m$, $n>n_{0}$
\begin{equation} \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}
\end{equation}
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
\begin{equation} \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}
\end{equation}
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
\begin{equation}  \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},
\end{equation}
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{equation}
\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}
\end{equation}
\begin{equation}
\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}
\end{equation}
By the dominated convergence theorem we have
\begin{equation} \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}
\end{equation}
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{equation}
\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}
\end{equation}
\begin{equation}
\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}
\end{equation}
 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.


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\end{thebibliography}

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