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

Let u be a type I blowing up solution of the Cauchy-Dirichlet problem for a semilinear heat equation, { ∂_tu = Δu + u^p, x ε Ω, t > 0, u(x, t) = 0, x ε ∂Ω, t > 0, u(x, 0) = φ(x), x ε Ω, where Ω is a (possibly unbounded) domain in R^N, N≥1, and p>1. We prove that, if φ ε L ^∞(Ω)∩L^q(Ω) for some q ε [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.