### 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 |
---|---|

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.

**Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations.** / Kaneko, Yuki; Oeda, Kazuhiro; Yamada, Yoshio.

Research output: Contribution to journal › Article

*Funkcialaj Ekvacioj*, vol. 57, no. 3, pp. 449-465.

}

TY - JOUR

T1 - Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations

AU - Kaneko, Yuki

AU - Oeda, Kazuhiro

AU - Yamada, Yoshio

PY - 2015/1/10

Y1 - 2015/1/10

N2 - 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.

AB - 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.

KW - Comparison principle

KW - Free boundary problem

KW - Reaction-diffusion equation

KW - Spreading and vanishing of species

KW - Upper and lower solutions

UR - http://www.scopus.com/inward/record.url?scp=84920996986&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84920996986&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84920996986

VL - 57

SP - 449

EP - 465

JO - Funkcialaj Ekvacioj

JF - Funkcialaj Ekvacioj

SN - 0532-8721

IS - 3

ER -