Blow-up set for a semilinear heat equation with small diffusion

We consider the blow-up problem for a semilinear heat equation, {partial derivative(t)u = epsilon Delta u + u(p) in Omega x (0, T), u(x, t) = 0 on partial derivative Omega x (0, T) if partial derivative Omega not equal phi, u(x, 0) = phi(epsilon)(x) >= 0 in Omega, where Omega is a domain in R-N, N >= 1, epsilon > 0, p > 1, and T > 0. In this paper, under suitable assumptions on {phi(epsilon)}, we prove that, if the family of the solutions {u(epsilon)} kid satisfies a uniform type I blow-up estimate with respect to epsilon, then the solution u(epsilon) blows up only near the maximum points of the initial datum phi(epsilon) for any sufficiently small epsilon > 0. This is proved without any conditions on the exponent p and the domain Omega, such as (N - 2)p < N + 2 and the convexity of the domain Omega. (C) 2010 Elsevier Inc. All rights reserved.