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

TL;DR

This paper investigates a free boundary problem for the Fisher-KPP equation with a moving boundary, establishing a trichotomy of possible long-term behaviors—vanishing, spreading, or transition—based on initial conditions and boundary dynamics.

Contribution

It introduces a novel free boundary model with a hostile left boundary moving at constant speed and classifies all possible solution behaviors into three categories.

Findings

Established a trichotomy of solution behaviors: vanishing, spreading, and transition.

Connected the results to similar models with shifting environments, highlighting differences in vanishing.

Provided conditions under which each of the three behaviors occurs.

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 behaviours, vanishing, spreading and transition, happens. This result is related to the results appears in the free boundary problem for the Fisher-KPP equation with a shifting-environment, which was considered by Du, Wei and Zhou [11](arXiv:1508.06246). However the vanishing in our problem is different from that in [11] because in our vanishing case, the solution is not global-in-time.