Uniform blow-up profiles for nonlinear and nonlocal reaction-diffusion equations

In this paper. we investigate the blow-up properties of positive solutions to the following integro-parabolic equations u(t) = Delta u + e(alpha u) + gamma - 1/vertical bar Omega vertical bar integral(Omega)e(beta u)dx, where alpha, beta > 0 and gamma > 1, and Omega = {x is an element of R-n :vertical bar x vertical bar < R}. For the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters alpha and beta. Moreover, the blow-up rates are also determined in each case. Particularly, for the special case: alpha = beta and n <= 2, it will be proved that the blow-up rate at x = 0 is faster than that at the other point x not equal 0. This seems to be a new phenomenon for this kind of problem. Crown Copyright (C) 2008 Published by Elsevier Ltd. All rights reserved.