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,equation(P){∂tu=Δu+up,x∈Ω,t>0,u(x,t)=0,x∈∂Ω,t>0,u(x,0)=φ(x),x∈Ω, where Ω   is a (possibly unbounded) domain in RNRN, N⩾1N⩾1, and p>1p>1. We prove that, if φ∈L∞(Ω)∩Lq(Ω)φ∈L∞(Ω)∩Lq(Ω) for some q∈[1,∞)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.