### Abstract

We will study a free boundary problem of the nonlinear diffusion equations of the form u_{t}=u_{xx}+f(u),t>0,ct<x<h(t), where f is C^{1} function satisfying f(0)=0, c>0 is a given constant and h(t) is a free boundary which is determined by a Stefan-like condition. This model may be used to describe the spreading of a new or invasive species with population density u(t,x) over a one dimensional habitat. The free boundary x=h(t) represents the spreading front. In this model, we impose zero Dirichlet boundary condition at left moving boundary x=ct. This means that the left boundary of the habitat is a very hostile environment for the species and that the habitat is eroded away by the left moving boundary at constant speed c. In this paper we will extend the results of a trichotomy result obtained in [23] to general monostable, bistable and combustion types of nonlinearities. We show that the long-time dynamical behavior of solutions can be expressed by unified fashion, that is, for any initial data, the unique solution exhibits exactly one of the behaviors, spreading, vanishing and transition. We also give the asymptotic profile of the solution over the whole domain when spreading happens. The approach here is quite different from that used in [23].

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

Pages (from-to) | 1000-1043 |

Number of pages | 44 |

Journal | Journal of Differential Equations |

Volume | 265 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 Aug 5 |

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### Keywords

- Bistable
- Combustion
- Free boundary problem
- Monostable
- Nonlinear diffusion equation

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*265*(3), 1000-1043. https://doi.org/10.1016/j.jde.2018.03.026

**Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary.** / Kaneko, Yuki; Matsuzawa, Hiroshi.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 265, no. 3, pp. 1000-1043. https://doi.org/10.1016/j.jde.2018.03.026

}

TY - JOUR

T1 - Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary

AU - Kaneko, Yuki

AU - Matsuzawa, Hiroshi

PY - 2018/8/5

Y1 - 2018/8/5

N2 - We will study a free boundary problem of the nonlinear diffusion equations of the form ut=uxx+f(u),t>0,ct1 function satisfying f(0)=0, c>0 is a given constant and h(t) is a free boundary which is determined by a Stefan-like condition. This model may be used to describe the spreading of a new or invasive species with population density u(t,x) over a one dimensional habitat. The free boundary x=h(t) represents the spreading front. In this model, we impose zero Dirichlet boundary condition at left moving boundary x=ct. This means that the left boundary of the habitat is a very hostile environment for the species and that the habitat is eroded away by the left moving boundary at constant speed c. In this paper we will extend the results of a trichotomy result obtained in [23] to general monostable, bistable and combustion types of nonlinearities. We show that the long-time dynamical behavior of solutions can be expressed by unified fashion, that is, for any initial data, the unique solution exhibits exactly one of the behaviors, spreading, vanishing and transition. We also give the asymptotic profile of the solution over the whole domain when spreading happens. The approach here is quite different from that used in [23].

AB - We will study a free boundary problem of the nonlinear diffusion equations of the form ut=uxx+f(u),t>0,ct1 function satisfying f(0)=0, c>0 is a given constant and h(t) is a free boundary which is determined by a Stefan-like condition. This model may be used to describe the spreading of a new or invasive species with population density u(t,x) over a one dimensional habitat. The free boundary x=h(t) represents the spreading front. In this model, we impose zero Dirichlet boundary condition at left moving boundary x=ct. This means that the left boundary of the habitat is a very hostile environment for the species and that the habitat is eroded away by the left moving boundary at constant speed c. In this paper we will extend the results of a trichotomy result obtained in [23] to general monostable, bistable and combustion types of nonlinearities. We show that the long-time dynamical behavior of solutions can be expressed by unified fashion, that is, for any initial data, the unique solution exhibits exactly one of the behaviors, spreading, vanishing and transition. We also give the asymptotic profile of the solution over the whole domain when spreading happens. The approach here is quite different from that used in [23].

KW - Bistable

KW - Combustion

KW - Free boundary problem

KW - Monostable

KW - Nonlinear diffusion equation

UR - http://www.scopus.com/inward/record.url?scp=85044791146&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044791146&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2018.03.026

DO - 10.1016/j.jde.2018.03.026

M3 - Article

VL - 265

SP - 1000

EP - 1043

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 3

ER -