Entire solutions in the Fisher-KPP equation with nonlocal dispersal

This paper is concerned with entire solutions of the Fisher-KPP equation with nonlocal dispersal, i.e., ut=J∗u−u+f(u)ut=J∗u−u+f(u), which is a one-dimensional nonlocal version of the Fisher-KPP equation describing the spatial spread of a mutant in a given population and the dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modeled by a convolution operator. Here the entire solutions are defined in the whole space and for all time t∈Rt∈R. A comparison principle is employed to establish the existence of entire solutions by combining two traveling wave solutions with different speeds and coming from both ends of the real axis and some spatially independent solutions. The main difficulty is that a lack of regularizing effect occurs. This is probably the first time the existence of entire solutions of reaction equations with nonlocal dispersal has been studied.