Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term

In this paper we study the blow-up problem for a non-local diffusion equation with a reaction term, u(t)(x,t) = integral(Omega)J(x - y)(u(y, t) - u(x, t))dy + u(p)(x, t). We prove that non-negative and non-trivial Solutions blow Lip in finite time if and only if p > 1. Moreover, we find that the blow-up rate is the same as the one that holds for the ODE u(t) = u(p), that is, lim(t NE arrow T)(T - t)(1/p-1)parallel to u(., t)parallel to(infinity) = (1/p-1)(1/p-1). Next, we deal with the blow-up set. We prove single point blow-up for radially symmetric solutions with a single maximum at the origin. as well as the localization of the blow-up set near any prescribed point, for certain initial conditions in a general domain with p > 2. Finally, we show some numerical experiments which illustrate our results. (C) 2008 Elsevier Ltd. All rights reserved.