A free boundary problem for the Fisher-KPP equation with a given moving boundary

Hiroshi Matsuzawa

doi:10.3934/cpaa.2018087

Article Contents

2018, Volume 17, Issue 5: 1821-1852. Doi: 10.3934/cpaa.2018087

This issue Previous Article Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy Next Article Global Carleman estimate for the Kawahara equation and its applications

A free boundary problem for the Fisher-KPP equation with a given moving boundary

Hiroshi Matsuzawa

National Institute of Technology, Numazu College, 3600 Ooka, Numazu City, Shizuoka 410-8501, Japan

Received: July 2017

Revised: December 2017

Published: April 2018

The author was partly supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) 17K05340.

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Abstract

We study free boundary problem of Fisher-KPP equation $u_t = u_{xx}+u(1-u), \ t>0, \ ct<x<h(t)$ . The number $c>0$ is a given constant, $h(t)$ is a free boundary which is determined by the Stefan-like condition. This model may be used to describe the spreading of a non-native species over a one dimensional habitat. The free boundary $x = h(t)$ represents the spreading front. In this model, we impose zero Dirichlet condition at left moving boundary $x = ct$ . This means that the left boundary of the habitat is a very hostile environment and that the habitat is eroded away by the left moving boundary at constant speed $c$ .

In this paper we will give a trichotomy result, that is, for any initial data, exactly one of the three behaviors, vanishing, spreading and transition, happens. This result is related to the results appear in the free boundary problem for the Fisher-KPP equation with a shifting-environment, which was considered by Du, Wei and Zhou [11]. However the vanishing in our problem is different from that in [11] because in our vanishing case, the solution is not global-in-time.

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Mathematics Subject Classification: Primary: 35R35, 35K20; Secondary: 35K55, 35K57, 35K58.

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