IUMJ

Title: Blow-up set for a semilinear heat equation and pointedness of the initial data

Authors: Kazuhiro Ishige and Yohei Fujishima

Issue: Volume 61 (2012), Issue 2, 627-663

Abstract:

We consider the blow-up problem for a semilinear heat equation, \begin{equation}\label{eq:E}\tag{E} \begin{cases} \partial_tu = \epsilon\Delta u + u^p, & x\in\Omega, t > 0,\\ u(x,t) = 0, & x\in\partial\Omega, t > 0 \mbox{ if }\partial\Omega \not= \emptyset,\\ u(x,0) = \varphi(x) \ge 0 (\not\equiv 0), & x\in\Omega, \end{cases} \end{equation} where $\epsilon > 0$, $p > 1$, $N \ge 1$, $\Omega$ is a domain in $\mathbb{R}^N$, and $\varphi$ is a nonnegative smooth bounded function in $\Omega$. It is known that, under suitable assumptions, if $\epsilon$ is sufficiently small, then the solution of \eqref{eq:E} blows up only near the maximum points of the initial function $\varphi$ (see, for example, [Y. Fujishima and K. Ishige, \textit{Blow-up set for a semilinear heat equation with small diffusion}, J. Differential Equations \textbf{249} (2010), no. 5, 1056--1077]). In this paper, as a continuation of [ibid.], we study the relationship between the location of the blow-up set and the level sets of the initial function $\varphi$. We also prove that the location of the blow-up set depends on the mean curvature of the graph of the initial function on its maximum points.