Spreading and vanishing in nonlinear diffusion problems with free boundaries

Abstract

We study nonlinear diffusion problems of the form ut = uxx + 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 Ϲ¹ and satisfies f (0) = 0, we show that the omega limit set ω(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 σ in the initial data, we reveal a threshold value σ* such that spreading (limt→∞u = 1) happens when σ > σ*, vanishing (limt→∞u = 0) happens when σ < σ*, and at the threshold value σ*, ω(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.