Please use this identifier to cite or link to this item: https://hdl.handle.net/1959.11/18389
Title: Spreading and vanishing in nonlinear diffusion problems with free boundaries
Contributor(s): Du, Yihong  (author); Lou, Bendong (author)
Publication Date: 2015
Open Access: Yes
DOI: 10.4171/JEMS/568Open Access Link
Handle Link: https://hdl.handle.net/1959.11/18389
Open Access Link: https://arxiv.org/abs/1301.5373Open Access Link
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.
Publication Type: Journal Article
Grant Details: ARC/DP120100727
Source of Publication: Journal of the European Mathematical Society, 17(10), p. 2673-2724
Publisher: European Mathematical Society Publishing House
Place of Publication: Zurich, Switzerland
ISSN: 1435-9855
1435-9863
Field of Research (FOR): 010110 Partial Differential Equations
Peer Reviewed: Yes
HERDC Category Description: C1 Refereed Article in a Scholarly Journal
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