Second critical exponent for a nonlinear nonlocal diffusion equation

This paper studies the Cauchy problem for a nonlocal diffusion equation ut=J∗u−u+up. We determine the second critical exponent, namely the optimal decay order for initial data at space infinity to distinguish global and blow-up solutions in the “super Fujita” range. Similar to the critical Fujita exponent obtained very recently by Alfaro (2017), we find that the second critical exponent also relies heavily on the behavior of the Fourier transform of the kernel function J.