Please use this identifier to cite or link to this item:
https://hdl.handle.net/1959.11/7468
Title: | Convergence and sharp thresholds for propagation in nonlinear diffusion problems | Contributor(s): | Du, Yihong (author) ; Matano, Hiroshi (author) | Publication Date: | 2010 | DOI: | 10.4171/JEMS/198 | Handle Link: | https://hdl.handle.net/1959.11/7468 | Abstract: | We study the Cauchy problem ut = uxx + f(u) (t > 0, x ∈ ℝ¹), u(0, x) = uₒ(x) (x ∈ ℝ¹), where f(u) is a locally Lipschitz continuous function satisfying f(0) = 0. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as t → ∞. Moreover, the limit is either a constant or a symmetrically decreasing stationary solution. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solution uλ, we show the existence of a sharp threshold between extinction (i.e., convergence to 0) and propagation (i.e., convergence to 1). The result holds even if f has a jumping discontinuity at u = 1. | Publication Type: | Journal Article | Source of Publication: | Journal of the European Mathematical Society, 12(2), p. 279-312 | Publisher: | European Mathematical Society Publishing House | Place of Publication: | Switzerland | ISSN: | 1435-9863 1435-9855 |
Fields of Research (FoR) 2008: | 010110 Partial Differential Equations | Socio-Economic Objective (SEO) 2008: | 970101 Expanding Knowledge in the Mathematical Sciences | Peer Reviewed: | Yes | HERDC Category Description: | C1 Refereed Article in a Scholarly Journal |
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Appears in Collections: | Journal Article School of Science and Technology |
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