The Stefan problem for the Fisher-KPP equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:253

Article

Du, Yihong¹  Guo, Zongming² 

[1] Univ New England, Sch Sci & Technol, Armidale, NSW 2351, Australia

[2] Henan Normal Univ, Dept Math, Xinxiang 453007, Peoples R China

关键词: Free boundary;    Stefan problem;    Spreading speed;    Fisher-KPP equation;   

DOI  :  10.1016/j.jde.2012.04.014

来源: Elsevier

【 摘 要 】

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. (c) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free