Electron. J. Differential Equations, Vol. 2020 (2020), No. 84, pp. 1-23.

Spatial dynamics of a nonlocal bistable reaction diffusion equation

Bang-Sheng Han, Meng-Xue Chang, Yinghui Yang

Abstract:
This article concerns a nonlocal bistable reaction-diffusion equation with an integral term. By using Leray-Schauder degree theory, the shift functions and Harnack inequality, we prove the existence of a traveling wave solution connecting 0 to an unknown positive steady state when the support of the integral is not small. Furthermore, for a specific kernel function, the stability of positive equilibrium is studied and some numerical simulations are given to show that the unknown positive steady state may be a periodic steady state. Finally, we demonstrate the periodic steady state indeed exists, using a center manifold theorem.

Submitted October 31, 2019. Published July 30, 2020.
Math Subject Classifications: 35C07, 35B40, 35K57, 92D25.
Key Words: Reaction-diffusion equation; traveling waves; numerical simulation; critical exponent.
DOI: 10.58997/ejde.2020.84

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Bang-Sheng Han
School of Mathematics
Southwest Jiaotong University
Chengdu, Sichuan, 611756, China
email: hanbangsheng@swjtu.edu.cn
Meng-Xue Chang
School of Mathematics
Southwest Jiaotong University
Chengdu, Sichuan, 611756, China
email: mengxue_chang@163.com
Yinghui Yang
School of Mathematics
Southwest Jiaotong University
Chengdu, Sichuan, 611756, China
email: yangyh8605@swjtu.edu.cn

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