Large time behaviour in nonlocal reaction-diffusion equations of the Fisher-KPP type

Jean-Michel Roquejoffre

Published 2022-04-26Version 1

The basic question is the evolution of the solutions to equations of the Fisher-KPP type, in which the diffusion is given by an integral operator. The level sets will organize themselves into an invasion front that is asymptotically linear in time, corrected by a logarithmic term. For a special class of nonlinearities, this fact can be deduced from the study of an underlying branching random walk (A\"idekon, 2013). This extends a famous result by Bramson (1983), where the diffusion is given by the Laplacian, and the underlying random walk is the Branching Brownian Motion. Motivated by models arising from epidemiology and ecology, that go beyond the equations arising from the branching random walks, this work explains how this logarithmic correction can be derived by working directly on the integro-differential equation.