Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity
Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity
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DOI：
10.3934/dcds.2020033
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Maho Endo;Y. Kaneko;Yoshio Yamada
中科院分区：
文献类型：
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作者：
Maho Endo;Y. Kaneko;Yoshio Yamada
本文讨论了一维区间内反应扩散方程的自由边界问题，其边界由固定端点和运动端点组成。我们把齐次Dirichlet条件放在固定边界上，而我们假设运动边界的动力学受Stefan条件支配。这类自由边界问题已被许多研究者研究过。我们将采用一个正双稳型的非线性反应项，它表现出解的有趣性质，如多重扩散现象。事实上，我们将证明解的大时间行为可以分为三种类型：消失、小扩散和大扩散。文中还给出了这些行为的充分条件。此外，对于两种类型的扩散，我们将给出每个自由边界的扩散速度和每个解的渐近轮廓的精确估计。
This paper deals with a free boundary problem for a reaction-diffusion equation in a one-dimensional interval whose boundary consists of a fixed end-point and a moving one. We put homogeneous Dirichlet condition at the fixed boundary, while we assume that the dynamics of the moving boundary is governed by the Stefan condition. Such free boundary problems have been studied by a lot of researchers. We will take a nonlinear reaction term of positive bistable type which exhibits interesting properties of solutions such as multiple spreading phenomena. In fact, it will be proved that large-time behaviors of solutions can be classified into three types; vanishing, small spreading and big spreading. Some sufficient conditions for these behaviors are also shown. Moreover, for two types of spreading, we will give sharp estimates of spreading speed of each free boundary and asymptotic profiles of each solution.