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https://hdl.handle.net/1959.11/12057
Title: | The Stefan problem for the Fisher-KPP equation | Contributor(s): | Du, Yihong (author) ; Guo, Zongming (author) | Publication Date: | 2012 | Open Access: | Yes | DOI: | 10.1016/j.jde.2012.04.014 | Handle Link: | https://hdl.handle.net/1959.11/12057 | Abstract: | We study the Fisher-KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the 'weak solution' for a class of free boundary problems that include the Fisher-KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson-Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here. | Publication Type: | Journal Article | Grant Details: | ARC/DP120100727 | Source of Publication: | Journal of Differential Equations, 253(3), p. 996-1035 | Publisher: | Academic Press | Place of Publication: | United States of America | ISSN: | 1090-2732 0022-0396 |
Fields of Research (FoR) 2008: | 010110 Partial Differential Equations | Fields of Research (FoR) 2020: | 490410 Partial differential equations | Socio-Economic Objective (SEO) 2008: | 970101 Expanding Knowledge in the Mathematical Sciences | Socio-Economic Objective (SEO) 2020: | 280118 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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