Differential and Integral Equations

Location of the blow-up set for a superlinear heat equation with small diffusion

Yohei Fujishima

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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.

Article information

Differential Integral Equations, Volume 25, Number 7/8 (2012), 759-786.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B44: Blow-up 35K20: Initial-boundary value problems for second-order parabolic equations 35K91: Semilinear parabolic equations with Laplacian, bi-Laplacian or poly- Laplacian


Fujishima, Yohei. Location of the blow-up set for a superlinear heat equation with small diffusion. Differential Integral Equations 25 (2012), no. 7/8, 759--786. https://projecteuclid.org/euclid.die/1356012662

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