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

Fujishima, Yohei; Ishige, Kazuhiro

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 231-247.

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

$$\begin{array}{cc}\{\begin{array}{cc}{\partial }_{t}u=\Delta u+u^p,\hfill & x\in \Omega ,t>0,\hfill \\ u(x,t)=0,\hfill & x\in \partial \Omega ,t>0,\hfill \\ u(x,0)=\varphi \left(x\right),\hfill & x\in \Omega ,\hfill \end{array}& \text{(P)}\end{array}$$

where Ω is a (possibly unbounded) domain in ${𝐑}^N$, $N⩾1$, and $p>1$. We prove that, if $\varphi \in L^{\infty }\left(\Omega \right)\cap L^q\left(\Omega \right)$ 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 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.

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