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, ut(x,t)=∫ΩJ(x−y)(u(y,t)−u(x,t))dy+up(x,t). We prove that non-negative and non-trivial solutions blow up in finite time if and only if p>1p>1. Moreover, we find that the blow-up rate is the same as the one that holds for the ODE ut=uput=up, that is, limt↗T(T−t)1p−1‖u(⋅,t)‖∞=(1p−1)1p−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>2p>2. Finally, we show some numerical experiments which illustrate our results.