Spreading and vanishing in nonlinear diffusion problems with free boundaries

Abstract

We study nonlinear diffusion problems of the form ut D uxx C f .u/ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f .u/ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f .u which is C1 and satisfies f .0/ D 0, we show that the omega limit set u of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-Time dynamical behavior of the problem; moreover, by introducing a parameter δ in the initial data, we reveal a threshold value δ∗ such that spreading (limt→∞1u D 1) happens when δ > , vanishing (limt!1u D 0) happens when δ > ω, and at the threshold value u is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.