Blow-up problems for a semilinear heat equation with large diffusion

We consider the blow-up problem of a semilinear heat equation, u(t)=D Delta u + u(p) in Omega x (0, T-D)(,) du/dv (x, t) = 0 on rho Omega x (0, T-D), u(x, 0) = phi(x) >= 0 in Omega, where Omega is a bounded smooth domain in R-N, T-D > 0, D > 0, and p > 1. We study the blowup time, the location of the blow-up set, and the blow-up profile of the blow-up solution for sufficiently large D. In particular, we prove that, for almost all initial data 0, if D is sufficiently large, then the solution blows-up only near the maximum points of the orthogonal projection of the initial data phi from L-2(Omega) onto the second Neumann eigenspace. (c) 2004 Elsevier Inc. All rights reserved.