The spreading speed of solutions of the non-local Fisher–KPP equation

We consider the Fisher–KPP equation with a non-local interaction term. In [16], Hamel and Ryzhik showed that in solutions of this equation, the front location at a large time t is 2t+o(t). We study the asymptotics of the second order term in the front location. If the interaction kernel ϕ(x) decays sufficiently fast as x→∞ then this term is given by −322log⁡t+o(log⁡t), which is the same correction as found by Bramson in [7] for the local Fisher–KPP equation. However, if ϕ has a heavier tail then the second order term is −tβ+o(1), where β∈(0,1) depends on the tail of ϕ. The proofs are probabilistic, using a Feynman–Kac formula. Since solutions of the non-local Fisher–KPP equation do not obey the maximum principle, the proofs differ from those in [7], although some of the ideas used are similar.