MR3308703 Yuki KANEKO, Kazuhiro OEDA and Yoshio YAMADA Yuki KANEKO, Kazuhiro OEDA and Yoshio YAMADA Funkcialaj Ekvacioj. Serio Internacia 57 2014 449--465 http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-3/57_449.pdf http://www.ams.org/mathscinet-getitem?mr=MR3308703 We discuss a free boundary problem for a diffusion equation in a one-dimensional interval which models the spreading of invasive or new species. Moreover, the free boundary represents a spreading front of the species and its dynamical behavior is determined by a Stefan-like condition. This problem has been proposed by Du and Lin (2010) and, recently, Kaneko and Yamada have studied a free boundary problem for a general reaction-diffusion equation under Dirichlet boundary conditions. The main purpose of this paper is to define “spreading” and “vanishing” of species for a free boundary problem with general nonlinearity and study the underlying principle to determine the spreading or vanishing behavior as time tends to infinity. It will be proved that vanishing occurs if and only if the free boundary stays in a bounded interval, and that, when vanishing occurs, the population decreases exponentially to zero in large time. Free boundary problem, Reaction-diffusion equation, Spreading and vanishing of species, Comparison principle, Upper and lower solutions. 35R35, 35K57, 92D25, 35J60. 57-449 2014 Yuki KANEKO, Kazuhiro OEDA and Yoshio YAMADA Yuki KANEKO, Kazuhiro OEDA and Yoshio YAMADA 1 Cantrell, R. S.; Cosner, C. Spatial Ecology via Reaction-Diffusion Equations Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester 2003 MR2191264

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