Propagation, diffusion and free boundaries

Abstract

In this short review, we describe some recent developments on the modelling of propagation by nonlinear partial differential equations, which involve local as well as nonlocal diffusion, and free boundaries. After a brief account of the classical works of Fisher, Kolmogorov–Petrovski–Piskunov (KPP), Skallem and Aronson-Weinberger, on the use of reaction-diffusion equations to model propagation and spreading speed, various models involving a free boundary are considered, which have the advantage of providing a clear spreading front over the classical models, apart from giving a spreading speed. These include nonlinear Stefan problems, the porous medium equation with a nonlinear source term, and nonlocal versions of the nonlinear Stefan problems in space dimension 1. The results selected here are mainly from recent works of the author and his collaborators, and care is taken to make the content accessible to readers who are not necessarily specialists in the area of the considered topics.