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

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.