Blow-up set for type I blowing up solutions for a semilinear heat equation

Abstract

Let $u$ be a type I blowing up solution of the Cauchy–Dirichlet problem for a semilinear heat equation,

where $\Omega$ is a (possibly unbounded) domain in ${\mathbf{R}}^N$, $N\ge 1$, and $p>1$. We prove that, if $\phi \in L^{\infty }(\Omega )\cap L^q(\Omega )$ for some $q\in [1,\infty )$, then the blow-up set of the solution u is bounded. Furthermore, we give a sufficient condition for type I blowing up solutions not to blow up on the boundary of the domain $\Omega$. This enables us to prove that, if $\Omega$ is an annulus, then the radially symmetric solutions of ($P$) do not blow up on the boundary $\partial \Omega$.