The Bramson delay in the non-local Fisher-KPP equation

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either 2t− (3/2) log t+O(1), as in the local case, or 2t−O(t) for some explicit β ∈ (0, 1). Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any β ∈ (0, 1), examples of Fisher-KPP type non-linearities fβ such that the front for the local Fisher-KPP equation with reaction term fβ is at 2t−O(t). Key-Words: Reaction-diffusion equations, Logarithmic delay, Parabolic Harnack inequality AMS Class. No: 35K57, 35Q92, 45K05, 35C07