### Abstract

We discuss a free boundary problem for a di¤usion equation in a onedimensional 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-di¤usion 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.

Original language | English |
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Pages (from-to) | 449-465 |

Number of pages | 17 |

Journal | Funkcialaj Ekvacioj |

Volume | 57 |

Issue number | 3 |

Publication status | Published - 2015 Jan 10 |

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### Keywords

- Comparison principle
- Free boundary problem
- Reaction-diffusion equation
- Spreading and vanishing of species
- Upper and lower solutions

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Geometry and Topology

### Cite this

*Funkcialaj Ekvacioj*,

*57*(3), 449-465.