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

We consider the nonlocal diffusion equation u t - u xx = u p (t,x 0 (t)), on the space interval (0,1), with Dirichlet boundary conditions. It is known that if the curve x 0 (t) remains in a compact subset of (0,1) for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on x 0 is sharp: for a large class of functions x 0 (t) approaching the boundary as t → ∞, 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.