We introduce and discuss a class of free boundary problems with "nonlocal diffusion", which are natural extensions of the free boundary models considered by Du and Lin [SIAM J. Math. Anal., 2010] and elsewhere, where "local diffusion" was used to describe the dispersal of the species being modeled, with the free boundary representing the spreading front of the species.
We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type.
We demonstrate that a spreading-vanishing dichotomy holds, though the spreading-vanishing criteria differ significantly from the well-known local diffusion model.