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July/August 2012 Location of the blow-up set for a superlinear heat equation with small diffusion
Yohei Fujishima
Differential Integral Equations 25(7/8): 759-786 (July/August 2012).

Abstract

We consider the blow-up problem for a superlinear heat equation $$ \begin{cases} \partial_t u=\epsilon\Delta u+f(u), &x\in\Omega, \,\,\, t>0, \\ u(x,t)=0, &x\in\partial\Omega, \,\,\, t>0 \quad\mbox{if}\quad \partial\Omega\not=\emptyset, \\ u(x,0)=\varphi_\epsilon(x)\ge 0\, (\not\equiv 0), &x\in\Omega, \end{cases} $$ where $\epsilon>0$, $N\ge 1$, $\Omega$ is a domain in ${\bf R}^N$, $f=f(s)$ is a convex function in $s\in (0,\infty)$, and the initial function $\varphi_\epsilon$ is a nonnegative bounded continuous function in $\overline{\Omega}$. The typical examples of $f$ that we treat in this paper, are $f(u)=(u+\lambda)^p$ ($p>1$, $\lambda\ge 0$) and $f(u)=e^u$. In this paper, under suitable assumptions, we prove that the solution $u_\epsilon$ blows up only near the maximum points of the initial function $\varphi_\epsilon$ if $\epsilon>0$ is sufficiently small.

Citation

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Yohei Fujishima. "Location of the blow-up set for a superlinear heat equation with small diffusion." Differential Integral Equations 25 (7/8) 759 - 786, July/August 2012.

Information

Published: July/August 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1265.35039
MathSciNet: MR2975694

Subjects:
Primary: 35B44, 35K20, 35K91

Rights: Copyright © 2012 Khayyam Publishing, Inc.

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Vol.25 • No. 7/8 • July/August 2012
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