Long-time dynamics of the Fisher-KPP equation with nonlocal diffusion and free boundary

Abstract

We consider the Fisher-KPP equation with free boundary and "nonlocal diffusion". We show the problem is well-posed, and its long-time dynamical behavior is governed by a spreading-vanishing dichotomy. Moreover, we completely determine the spreading profile, which may have a finite spreading speed determined by a semi-wave problem, or have infinite spreading speed (accelerated spreading), according to whether a threshold condition on the kernel function is satisfied. Further more, for some typical kernel functions, we obtain sharp estimates of the spreading speed (whether finite or infinite).