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

摘要

Let Ω be a bounded smooth domain in R~N. We consider the problem ut = Δu + V(x)u~p in Ω * [0,T), with Dirichlet boundary conditions u = 0 on (partail deriv)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 φ~(p-1)V attains its maximum.