Large time behaviour in nonlocal reaction-diffusion equations

The basic question is the evolution of the level sets of the solutions to equations of the Fisher-KPP type, in which the diffusion is given by an integral operator. Mathematically, they will organise 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 (Aïdekon, 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 a models arising from epidemiology and ecology, I will explain how the linear spreading and the logarithmic correction can be derived by working directly on the integro-differential equation. The inspiration comes from the situation where the diffusion is given by the Laplacian, but some important differences should be underlined.