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=Δu+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.