In this paper, we address the following initial value problem:

             in

             in

where Ω is a bounded domain in  with smooth boundary is a C¹ convex nondecreasing function,   is a kernel which is nonnegative and bounded in  Under some conditions, we show that the solution of a perturbed form of the above problem blows up in a finite time and estimate its blow-up time. We also prove the continuity of the blow-up time as a function of the initial datum. Finally, we give some numerical results to illustrate our analysis.

nonlocal diffusion, blow-up, numerical blow-up time, finite difference method, continuity.