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

Let a be a type I blowing up solution of the Cauchy -Dirichlet problem for a semilinear heat equation, { partial derivative(t)u = Delta u + u(P), x is an element of Omega, t > 0, u(x, t) = 0, x is an element of partial derivative Omega, t > 0, u (x, 0) = phi(x), x is an element of Omega, where Omega is a (possibly unbounded) domain in R-N, N >= 1, and p > 1. We prove that, if phi is an element of L-infinity (Omega) boolean AND L-q (Omega) for some q is an element of [1, infinity), 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 derivative Omega. (C) 2013 Elsevier Masson SAS. All rights reserved.