A free boundary problem for the Fisher-KPP equation with a given moving boundary
A free boundary problem for the Fisher-KPP equation with a given moving boundary
复制标题
DOI：
10.3934/cpaa.2018087
复制
发表时间：
2017-08
期刊：
影响因子：
--
通讯作者：
H. Matsuzawa
中科院分区：
文献类型：
--
作者：
H. Matsuzawa
研究了Fisher-KPP方程u_t=u_{xx}+u（1-u）的自由边值问题，\ t>0，\ ct 0$是一个给定的常数，h（t）$是一个由Stefan-like条件确定的自由边值问题.该模型可用于描述非本地物种在一维生境中的扩散。自由边界$x=h（t）$表示扩展前沿。在这个模型中，我们在左移动边界x=ct$处施加零Dirichlet条件.这意味着栖息地的左边界是一个非常恶劣的环境，栖息地被以恒定速度$c$的左移动边界侵蚀。本文将给出一个可分性结果，即对于任何初始数据，消失、扩散和跃迁三种行为中的一种恰好发生。这一结果与杜、魏、周[11]（arXiv：1508.06246）在迁移环境下Fisher-KPP方程自由边界问题中的结果有关。然而，我们的问题中的消失与[11]中的不同，因为在我们的消失情况下，解决方案不是全球性的。
We study free boundary problem of Fisher-KPP equation $u_t=u_{xx}+u(1-u),\ t>0,\ ct 0$ 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.