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

e 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 setΩ + := {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.  2004 Elsevier Inc. All rights reserved.