A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems

We study a priori estimates of positive solutions of the equation ∂tu−Δu=λu+a(x)up, x∈Ω, t>0, satisfying the homogeneous Dirichlet boundary conditions. Here Ω is a bounded domain in Rn, λ∈R, p>1 is subcritical, a∈C(Ω¯) changes sign and a,p satisfy some additional technical hypotheses. Assume that the solution u blows up in a finite time T and the set Ω+:={x∈Ω:a(x)>0} is connected. Using our a priori bounds, we show that u blows up completely in Ω+ at t=T and the blow-up time T depends continuously on the initial data.