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 $2t - ({3}/2)\log t + O(1)$,asinthelocalcase,or$2t - O(t^\beta)$ for some explicit $\beta \in (0,1)$. Our main tools here are alocal-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $\beta \in (0,1)$, examples of Fisher-KPP type non-linearities $f\mathrm{\_}\beta$ such that the front for the localFisher-KPP equation with reaction term $f\mathrm{\_}\beta$ is at $2t-O(t^{\beta })$.