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

Abstract

Let Omega be a bounded smooth domain in R-N. We consider the problem u(t) = Delta u + V(x)u(P) in Omega x [0, T), with Dirichlet boundary conditions u = 0 on partial derivative Omega x [0, T) and initial datum u (x, 0) = M phi (x) where M >= 0, phi 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 phi(P-1) V attains its maximum. (c) 2007 Elsevier Inc. All rights reserved.