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September 2018, 17(5): 1821-1852. doi: 10.3934/cpaa.2018087

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

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

Received  July 2017 Revised  December 2017 Published  April 2018

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

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.

Citation: Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
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