Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du ; Bendong Lou

Journal of the European Mathematical Society, Tome 17 (2015) no. 10, pp. 2673-2724.

Résumé

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 C1 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∗, 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.

Détail

DOI : 10.4171/jems/568

Classification : 35-XX
Keywords: Nonlinear diffusion equation, free boundary problem, asymptotic behavior, monostable, bistable, combustion, sharp threshold, spreading speed