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

### Yohei Fujishima

Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531, Japan### Kazuhiro Ishige

Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

## Abstract

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

$⎩⎨⎧ ∂_{t}u=Δu+u_{p},u(x,t)=0,u(x,0)=φ(x), x∈Ω,t>0,x∈∂Ω,t>0,x∈Ω, $

where $Ω$ is a (possibly unbounded) domain in $R_{N}$, $N≥1$, and $p>1$. We prove that, if $φ∈L_{∞}(Ω)∩L_{q}(Ω)$ for some $q∈[1,∞)$, 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 $Ω$. This enables us to prove that, if $Ω$ is an annulus, then the radially symmetric solutions of ($P$) do not blow up on the boundary $∂Ω$.

## Cite this article

Yohei Fujishima, Kazuhiro Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 2, pp. 231–247

DOI 10.1016/J.ANIHPC.2013.03.001