Infinite time blow-up for superlinear parabolic problems with localized reaction

We consider the nonlocal diffusion equation

on the space interval , with Dirichlet boundary conditions. It is known that if the curve remains in a compact subset of for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on is sharp: for a large class of functions approaching the boundary as , blow-up in infinite time does occur for certain initial data. Moreover, the asymptotic behavior of the corresponding solution is precisely estimated and more general nonlinearities are also considered.