### Abstract

This paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We will completely classify asymptotic behaviors of solutions and, in particular, observe two different types of spreading phenomena corresponding to two positive stable equilibrium states. Moreover, it will be proved that, if the free boundary expands to infinity, an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi-wave problem.

Original language | English |
---|---|

Pages (from-to) | 538-572 |

Number of pages | 35 |

Journal | Journal of Differential Equations |

Volume | 261 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Jul 5 |

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### Keywords

- Comparison theorem
- Free boundary problem
- Reaction-diffusion equation
- Semi-wave
- Spreading
- Vanishing

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*261*(1), 538-572. https://doi.org/10.1016/j.jde.2016.03.017

**Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity.** / Kawai, Yusuke; Yamada, Yoshio.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 261, no. 1, pp. 538-572. https://doi.org/10.1016/j.jde.2016.03.017

}

TY - JOUR

T1 - Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity

AU - Kawai, Yusuke

AU - Yamada, Yoshio

PY - 2016/7/5

Y1 - 2016/7/5

N2 - This paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We will completely classify asymptotic behaviors of solutions and, in particular, observe two different types of spreading phenomena corresponding to two positive stable equilibrium states. Moreover, it will be proved that, if the free boundary expands to infinity, an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi-wave problem.

AB - This paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We will completely classify asymptotic behaviors of solutions and, in particular, observe two different types of spreading phenomena corresponding to two positive stable equilibrium states. Moreover, it will be proved that, if the free boundary expands to infinity, an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi-wave problem.

KW - Comparison theorem

KW - Free boundary problem

KW - Reaction-diffusion equation

KW - Semi-wave

KW - Spreading

KW - Vanishing

UR - http://www.scopus.com/inward/record.url?scp=84961837637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84961837637&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2016.03.017

DO - 10.1016/j.jde.2016.03.017

M3 - Article

VL - 261

SP - 538

EP - 572

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -