\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 $$\label{e1.1} \begin{gathered} u_{1t}=a_1^2\Delta u_1 + f_1(x,t,u_1,\dots ,u_k), \\ \dots\\ u_{kt}=a_k^2\Delta u_k + f_k(x,t,u_1,\dots ,u_k), \end{gathered}$$ subject to the initial conditions $$\label{e1.2} u_{1}(x,0) =\varphi_1 (x), \dots, u_{k}(x,0) =\varphi_k (x),$$ where $x \in \mathbb{R}^n$, $n \geq 1$, $00$, $0p$ and $0\leq t\leq T$, then $$\label{e3.6} |u_{im} (x,t) - g_{im} (x,t)|<\delta.$$ \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 $0p_1$ 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{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} 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 $$\label{e3.9} |I_2| + |I_3| + |I_5| <\delta/2 \quad \text{if } |x|>p_2$$ 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 $$\label{e3.10} |I_1| + |I_4|<\delta/2 \quad \text{if } |x|>p_3$$ 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 $$\label{e3.11} |u_{im} (x,t) - g_i (t)| \leq |u_{im} (x,t) - g_{im} (t)| + |g_{im} (t) - g_{i} (t)| < \varepsilon, \quad i=1,\dots ,k,$$ 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} \begin{thebibliography}{99} \bibitem{BF} J. Bebernes, W. Fulks; The small heat-loss problem, \emph{J. Diff. Equat.} \textbf{57} (1985), 324-332. \bibitem{GU1} Y. Giga, N. Umeda; Blow-up directions at space infinity for solutions of semilinear heat equations, \emph{Bol. Soc. Parana Mat.} \textbf{23} (2005), 9-28. \bibitem{GU2} Y. Giga, N. Umeda; On blow-up at space infinity for semilinear heat equations, \emph{J. Math. Anal. Appl.} \textbf{316} (2006), 538-555. \bibitem{G} A. L. Gladkov; Behavior of solutions of semilinear parabolic equations, \emph{Matematicheskie Zametki} \textbf{51} (1992), 29-34 (in Russian). \bibitem{L} A. A. 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