Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity

Maho Endo; Yuki Kaneko; Yoshio Yamada

doi:10.3934/dcds.2020033

Article Contents

2020, Volume 40, Issue 6: 3375-3394. Doi: 10.3934/dcds.2020033 open access

This issue Previous Article Spike solutions for a mass conservation reaction-diffusion system Next Article Large time behavior of ODE type solutions to nonlinear diffusion equations

Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity

Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

^* Corresponding author: Yoshio Yamada
^* Corresponding author: Yoshio Yamada

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday
¹ Partially supported by Waseda University Grant for Special Research Project (2018K-203, 2018B-103) and Grant-in-Aid for Early-Career Scientists (19K14602).
² Partially supported by Grant-in-Aid for Scientific Research (C) 16K05244.
^** Current address: Department of Mathematical and Physical Sciences, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo 112-8681, Japan.

Received: December 31, 2018

Revised: July 31, 2019

Published: March 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 35R35; Secondary: 35K57, 35J61, 92D25.

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