The blow-up problem for a semilinear parabolic equation with a potential

Let Ω be a bounded smooth domain in RN. We consider the problem ut = u + V (x)up in Ω × [0,T ), with Dirichlet boundary conditions u = 0 on ∂Ω ×[0,T ) and initial datum u(x, 0) = Mϕ(x) where M 0, ϕ is positive and compatible with the boundary condition.We give estimates for the blow-up time of solutions for large values of M. As a consequence of these estimates we find that, for M large, the blow-up set concentrates near the points where ϕp−1V attains its maximum. © 2007 Elsevier Inc. All rights reserved.