\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 07, pp. 1--5.\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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/07\hfil Stabilization of solutions]
{Stabilization of solutions for semilinear parabolic systems as
$|x|\to \infty$}

\author[A. Gladkov \hfil EJDE-2009/07\hfilneg]
{Alexander Gladkov}

\address{Alexander Gladkov \hfill\break
Mathematics Department, Vitebsk State University, Moskovskii pr.
33, 210038 Vitebsk, Belarus}
\email{gladkoval@mail.ru}

\thanks{Submitted October 13, 2008. Published January 6, 2009.}
\subjclass[2000]{35K55, 35K65}
\keywords{Semilinear parabolic systems; stabilization}

\begin{abstract}
 We prove that solutions of the Cauchy problem for semilinear 
 parabolic systems converge to solutions of the Cauchy problem
 for a corresponding systems of ordinary differential equations,
 as $|x| \to \infty$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}\label{intro}

 In this paper we consider the Cauchy problem for the
system of semilinear parabolic equations
\begin{equation} \label{e1.1}
\begin{gathered}
u_{1t}=a_1^2\Delta u_1 + f_1(x,t,u_1,\dots ,u_k),  \\
\dots\\
u_{kt}=a_k^2\Delta u_k + f_k(x,t,u_1,\dots ,u_k),
\end{gathered}
\end{equation}
subject to the initial conditions
\begin{equation} \label{e1.2}
u_{1}(x,0) =\varphi_1 (x), \dots,
u_{k}(x,0) =\varphi_k (x),
\end{equation}
where $x \in \mathbb{R}^n$, $n \geq 1$, $0<t<T_0$, $T_0\leq
\infty$. Put $S_T = \mathbb{R}^n \times [0,T)$,
$\mathbb{R}^k_+=\{x\in \mathbb{R}^k: x_i \geq 0, i=1,\dots ,k \}$.
We assume that the data of problem \eqref{e1.1}-\eqref{e1.2}
satisfy the following conditions:
\begin{gather}
\parbox{.8\linewidth}{
$f_i(x,t,u_1,\dots ,u_k)$, $i=1,\dots ,k$ are defined and locally
H\"{o}lder continuous functions in $\mathbb{R}^n
\times[0,T_0)\times \mathbb{R}^k_+ $ and $\varphi_i(x)$,
$i=1,\dots ,k$ are continuous functions in $\mathbb{R}^n$;
}\label{e1.3}
\\
\label{e1.4}
f_i(x,t,u_1,\dots ,u_k), \quad i=1,\dots ,k \text{ do not
decrease in $u_1,\dots ,u_k$;}
\\
\parbox{.8\linewidth}{
$f_i(x,t,u_1,\dots ,u_k) \to \bar{f}_i(t,u_1,\dots ,u_k)$,
$i=1,\dots ,k$,  as $|x| \to \infty$ uniformly on any bounded
subset of $[0,T_0)\times \mathbb{R}^k_+$;}\label{e1.5}
\\
\label{e1.6}
0 \leq f_i(x,t,u_1,\dots ,u_k) \leq \bar{f}_i(t,u_1,\dots ,u_k),\quad
i=1,\dots ,k;
\\
 \label{e1.7}
0 \leq \varphi_i(x) \leq c_i, \quad
\lim_{|x| \to \infty }\varphi_i(x) = c_i, \quad
c_i \geq 0, \quad i=1,\dots ,k.
\end{gather}
The above assumptions  are satisfied, in
particular, for large class problems \eqref{e1.1}-\eqref{e1.2}, whose solutions exist only on a finite time
interval. Note also that the solution of \eqref{e1.1}-\eqref{e1.2} may not be unique.

Let us consider the Cauchy problem for the  system of
ordinary differential equations
\begin{equation} \label{e1.8}
\begin{gathered}
g'_{1}=\bar{f}_1(t,g_1,\dots ,g_k),  \\
\dots \\
g'_{k}=\bar{f}_k(t,g_1,\dots ,g_k),
\end{gathered}
\end{equation}
subject to the initial conditions
\begin{equation}
g_{1}(0) =c_1, \dots, g_{k}(0) =c_k.
 \label{e1.9}
\end{equation}
We suppose that the minimal nonnegative solution $g_i (t)$,
$i=1,\dots ,k$, of \eqref{e1.8}-\eqref{e1.9} exists on $[0,T_0)$.
The main result of the paper is the following theorem.

\begin{theorem}\label{Th1}
Let $u_i (x,t)$ be the minimal nonnegative
solution of the problem \eqref{e1.1}-\eqref{e1.2}. Then
\begin{equation}\label{e1.10}
u_i (x,t) \to g_i (t), \quad i=1,\dots ,k,  \quad \text{as }
|x| \to \infty
\end{equation}
uniformly for $t \in [0,T]$, $(T<T_0)$.
\end{theorem}

The behavior of solutions of parabolic equations as $|x| \to
\infty$ has been investigated by several authors. The case of one
semilinear parabolic equation on half line has been considered in
\cite{BF,L} for nonlinearities $f(x,t,u)=u^p$ and $f(x,t,u)=\exp
u$. The same problem with general nonlinearity $f(x,t,u)$ has been
investigated in \cite{G}. The behavior of solutions of nonlinear
parabolic equations for the Cauchy problem as $|x| \to \infty$ has
been analyzed in  \cite{GU1,GU2,S,SSU}.

The plan of this paper is as follows. In the next section, the
existence of a minimal solution for the problem \eqref{e1.1}-\eqref{e1.2}
is proved. The proof of Theorem~\ref{Th1} is given
in Section 3.

\section{Existence of a minimal solution}

We prove the existence of a minimal solution for
\eqref{e1.1}-\eqref{e1.2}. It is well known that
\eqref{e1.1}-\eqref{e1.2} is equivalent to the
system
\begin{equation} \label{e2.1}
\begin{gathered}
\begin{aligned}
u_{1}(x,t)
&= \int_{\mathbb{R}^n} E_1(x-y,t) \varphi_1 (y) \, dy \\
&\quad +\int_0^t\int_{\mathbb{R}^n} E_1(x-y,t-\tau) f_1(y,\tau,u_1,\dots ,u_k)
\, dy\,d\tau,  \end{aligned}\\
\dots  \\
\begin{aligned}
u_{k}(x,t)
&= \int_{\mathbb{R}^n} E_k(x-y,t) \varphi_k (y) \, dy \\
&\quad +\int_0^t\int_{\mathbb{R}^n} E_k(x-y,t-\tau)
f_k(y,\tau,u_1,\dots ,u_k) \, dy\,d\tau,
\end{aligned} \end{gathered}
\end{equation}
where $E_i(x,t)=(2a_i\sqrt{\pi t})^{-n} \exp(-|x|^2/[4a_i^2t])$,
$i=1,\dots ,k$, are the fundamental solutions of the correspondent heat
equations.

Let $u_{i0} (x,t) \equiv 0$, $i=1,\dots ,k$. We define sequences of
functions $u_{im} (x,t)$, $i=1,\dots ,k$, $m \in\mathbb{N}$,  the
following way
\begin{equation}\label{e2.2}
\begin{aligned}
u_{im}(x,t)&= \int_{\mathbb{R}^n} E_i(x-y,t) \varphi_i (y) \, dy \\
&\quad +\int_0^t\int_{\mathbb{R}^n} E_i(x-y,t-\tau)
f_i(y,\tau,u_{1(m-1)},\dots ,u_{k(m-1)}) \, dy\,d\tau.
\end{aligned}
\end{equation}
Obviously, the functions $g_i(t)$, $i=1,\dots ,k$, satisfy the integral
equations
\begin{equation}\label{e2.3}
g_{i}(t)= \int_{\mathbb{R}^n} E_i(x-y,t) c_i \, dy +
\int_0^t\int_{\mathbb{R}^n} E_i(x-y,t-\tau)
\bar{f}_i(\tau,g_{1},\dots ,g_{k}) \, dy\,d\tau.
\end{equation}
Using (\ref{e1.4}), (\ref{e1.6}), (\ref{e2.2}) and
(\ref{e2.3}), we have
\begin{equation}\label{e2.4}
0\leq u_{i(m-1)}(x,t) \leq u_{im}(x,t) \leq g_i(t),\quad
 i=1,\dots ,k,\;  m\in\mathbb{N}.
\end{equation}
By the Lebesgue theorem, and from (\ref{e2.2}) and (\ref{e2.4}),
we obtain that the sequences $u_{im}(x,t)$ converge to functions
$u_{i}(x,t)$ that satisfy (\ref{e2.1}), which means, ones satisfy
the problem \eqref{e1.1}-\eqref{e1.2}. Let $v_{i}(x,t)$,
$i=1,\dots ,k$, be any other solution of  \eqref{e1.1}-\eqref{e1.2}.
 By induction on $m$ it is easy to prove that
$u_{im}(x,t) \leq v_{i}(x,t)$, $i=1,\dots ,k$, $m\in\mathbb{N}$.
Therefore, $u_{i}(x,t)$, $i=1,\dots ,k$, is the minimal nonnegative
solution of this problem. We have proved the following statement.

\begin{theorem}\label{Th2}
There exists a minimal nonnegative solution
$u_{i}(x,t), i=1,\dots ,k$, of the problem \eqref{e1.1}-\eqref{e1.2} in $S_{T_0}$ that satisfies the inequalities
\begin{equation}\label{e2.5}
0 \leq u_{im} (x,t) \leq u_{i} (x,t) \leq g_i (t),  \quad
 (x,t)\in S_{T_0},\; i=1,\dots ,k,\; m \in\mathbb{N}.
\end{equation}
\end{theorem}

\section{Behavior of a minimal solution as $|x| \to \infty$}

We show that for the minimal nonnegative solution of
\eqref{e1.1}-\eqref{e1.2}, property  (\ref{e1.10}) is satisfied.
We define sequences of functions $ g_{im}(t)$, $i=1,\dots ,k$,
$m=0,1,...,$ as follows
\begin{equation}\label{e3.1}
 g_{i0}(t)\equiv 0, \quad
 g_{im}(t)= \int_0^t \bar{f}_i(\tau,g_{1(m-1)},\dots ,g_{k(m-1)}) \, d\tau
 + c_i, \quad
i=1,\dots ,k,\; m \in\mathbb{N}.
\end{equation}
Obviously, the sequences  $g_{im}(t)$ are monotonically nondecreasing,
converging to the minimal nonnegative solution $g_{i}(t)$,
$i=1,\dots ,k$, of problem
\eqref{e1.8}-\eqref{e1.9} on any interval $[0,T]$, $(T<T_0)$, and
\begin{equation}\label{e3.2}
 g_{im}(t)\leq g_{i}(t), \quad i=1,\dots ,k,\; m \in\mathbb{N}.
\end{equation}
According to the Dini criterion on uniform convergence of
functional sequences, we have
\begin{equation}\label{e3.3}
 g_{im}(t)\to g_{i}(t), \quad i=1,\dots ,k,  \text{ as }
 m \to \infty   \text{ uniformly on } [0,T].
\end{equation}
It is easy to prove that $g_{im}(t)$, $i=1,\dots ,k$,
$m\in\mathbb{N}$, satisfy the following equations
\begin{equation}\label{e3.4}
g_{im}(t)= \int_0^t\int_{\mathbb{R}^n} E_i(x-y,t-\tau)
\bar{f}_i(\tau,g_{1(m-1)},\dots ,g_{k(m-1)}) \, dy\,d\tau + c_i.
\end{equation}

Now we prove an auxiliary lemma.

\begin{lemma}\label{lem1}
For any $\delta>0$, $0<T<T_0$, $i=1,\dots ,k$,  and
$m\geq 0$ there exists a constant $p$ such
that if $|x|>p$ and $0\leq t\leq T$, then
\begin{equation}\label{e3.6}
|u_{im} (x,t) - g_{im} (x,t)|<\delta.
\end{equation}
\end{lemma}

\begin{proof}
 We use induction on $m$. It is
obviously that $u_{i0} (x,t) - g_{i0} (t)=0$, $i=1,\dots ,k$.
We assume that (\ref{e3.6}) holds for $m=l$, and we shall prove the
inequality for $m=l+1$. By the induction assumption, for any
$\varepsilon_1>0$ and $0<T<T_0$ there exists $p_1$ such that
\begin{equation}\label{e3.7}
|u_{il} (x,t) - g_{il} (t)|<\varepsilon_1,\quad i=1,\dots ,k,
\end{equation}
if $|x|>p_1$ and $0\leq t\leq T$.
Put $B(q)=\{ x \in \mathbb{R}^n: |x| \leq q \}$. From
(\ref{e2.2}) and (\ref{e3.4}), we have
\begin{equation}
\begin{aligned}
&|u_{i(l+1)} - g_{i(l+1)}| \\
&\leq \big|\int_0^t\int_{B(q)}
E_i(x-y,t-\tau) (f_i(y,\tau,u_{1l},\dots ,u_{kl}) -
\bar{f}_i(\tau,g_{1l},\dots ,g_{kl}))\, dy\,d\tau \big|\\
&\quad + \big|\int_0^t\int_{\mathbb{R}^n \backslash B(q)}
E_i(x-y,t-\tau) (f_i(y,\tau,u_{1l},\dots ,u_{kl}) -
\bar{f}_i(\tau,u_{1l},\dots ,u_{kl}))\, dy\,d\tau \big|\\
&\quad + \big|\int_0^t\int_{\mathbb{R}^n \backslash B(q)}
E_i(x-y,t-\tau) (\bar{f}_i(\tau,u_{1l},\dots ,u_{kl}) -
\bar{f}_i(\tau,g_{1l},\dots ,g_{kl}))\, dy\,d\tau \big|\\
&\quad + \big|\int_{B(q)} E_i(x-y,t) (\varphi_i(y) - c_i)\, dy \big| +
\big|\int_{\mathbb{R}^n \backslash B(q)} E_i(x-y,t) (\varphi_i(y)
- c_i)\, dy  \big|,
\end{aligned} \label{e3.8}
\end{equation}
where $q$ will be choose later. We denote by $I_j$, $j=1,\dots,5$
the integrals from the right-hand side of (\ref{e3.8}),
respectively. Obviously, $\bar{f}_i(t,u_1,\dots ,u_k)$, $i=1,\dots
,k$, are uniformly continuous on any compact subset of
$[0,T]\times \mathbb{R}^k_+$. Using this and (\ref{e1.5}),
(\ref{e1.7}), (\ref{e2.4}), (\ref{e3.2}), (\ref{e3.7}) for
suitable $\varepsilon_1$ and $q$, we get
\begin{equation}\label{e3.9}
|I_2| + |I_3| + |I_5| <\delta/2 \quad \text{if } |x|>p_2
\end{equation}
for some $p_2$. Since $E_i(x-y,t) \to 0$ as $|x| \to \infty$
uniformly on $[0,T]\times B(q)$, we have
\begin{equation}\label{e3.10}
|I_1| + |I_4|<\delta/2 \quad \text{if } |x|>p_3
\end{equation}
for some $p_3$. Now (\ref{e3.6}) follows from (\ref{e3.9}),
(\ref{e3.10}).
\end{proof}

\begin{proof}[Proof of Theorem \ref{Th1}]
We fix a positive $\varepsilon$. From Lemma \ref{lem1} and
(\ref{e3.3}),  for suitable $m$ and $q$, we have
\begin{equation}\label{e3.11}
|u_{im} (x,t) - g_i (t)| \leq |u_{im} (x,t) - g_{im} (t)| +
|g_{im} (t) - g_{i} (t)| < \varepsilon, \quad i=1,\dots ,k,
\end{equation}
if $|x|>q$ and $0 \leq t \leq T$. From (\ref{e2.5}) and
(\ref{e3.11}) we obtain
\begin{equation*}
g_{i}(t)-\varepsilon \leq u_{im} (x,t) \leq u_{i} (x,t) \leq
 g_{i} (t), \quad i=1,\dots ,k,
\end{equation*}
for $|x|>q$ and $0 \leq t \leq T$. The statement of the theorem
follows immediately from these arguments.
\end{proof}

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\end{document}