Global solutions for a semilinear heat equation in the exterior domain of a compact set

Kazuhiro Ishige; Michinori Ishiwata

doi:10.3934/dcds.2012.32.847

Article Contents

2012, Volume 32, Issue 3: 847-865. Doi: 10.3934/dcds.2012.32.847

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Global solutions for a semilinear heat equation in the exterior domain of a compact set

Kazuhiro Ishige^1, and
Michinori Ishiwata^2,

1.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578
2.
Faculty of Symbiotic Systems Science, Fukushima University, Kanayagawa, Fukushima 960-1269

Received: September 2010

Revised: February 2011

Published: March 2012

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Abstract

Let $u$ be a global in time solution of the Cauchy-Dirichlet problem for a semilinear heat equation, $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u+u^p,\quad & x\in\Omega,\,\, t>0,\\ u=0,\quad & x\in\partial\Omega,\,\,t>0,\\ u(x,0)=\phi(x)\ge 0,\quad & x\in\Omega, \end{array} \right. $$ where $\partial_t=\partial/\partial t$, $p>1+2/N$, $N\ge 3$, $\Omega$ is a smooth domain in ${\bf R}^N$, and $\phi\in L^\infty(\Omega)$. In this paper we give a sufficient condition for the solution $u$ to behave like $\|u(t)\|_{L^\infty({\bf R}^N)}=O(t^{-1/(p-1)})$ as $t\to\infty$, and give a classification of the large time behavior of the solution $u$.

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Mathematics Subject Classification: Primary: 35B40, 35K58; Secondary: 35B44.

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