Author(s) |
Du, Yihong
Lou, Bendong
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Publication Date |
2015
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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.
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Citation |
Journal of the European Mathematical Society, 17(10), p. 2673-2724
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ISSN |
1435-9863
1435-9855
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Link | |
Publisher |
European Mathematical Society Publishing House
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Title |
Spreading and vanishing in nonlinear diffusion problems with free boundaries
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Type of document |
Journal Article
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Entity Type |
Publication
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