Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect

Meng-Xue Chang; Bang-Sheng Han; Xiao-Ming Fan; Meng-Xue Chang; Bang-Sheng Han; Xiao-Ming Fan

doi:10.3934/era.2021024

Electronic Research Archive

2021, Volume 29, Issue 5: 3017-3030. doi: 10.3934/era.2021024

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Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

Received: 01 December 2020 Revised: 01 February 2021 Published: 16 March 2021
Primary: 35A01, 35K55, 35K57, 92D25; Secondary: 35K57

This paper is devoted to studying the Cauchy problem corresponding to the nonlocal bistable reaction diffusion equation. It is the first attempt to use the method of comparison principle to study the well-posedness for the nonlocal bistable reaction-diffusion equation. We show that the problem has a unique solution for any non-negative bounded initial value by using Gronwall's inequality. Moreover, the boundedness of the solution is obtained by means of the auxiliary problem. Finally, in the case that the initial data with compactly supported, we analyze the asymptotic behavior of the solution.
- Reaction-diffusion,
- nonlocal,
- existence,
- asymptotic
Citation: Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect[J]. Electronic Research Archive, 2021, 29(5): 3017-3030. doi: 10.3934/era.2021024

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Abstract

This paper is devoted to studying the Cauchy problem corresponding to the nonlocal bistable reaction diffusion equation. It is the first attempt to use the method of comparison principle to study the well-posedness for the nonlocal bistable reaction-diffusion equation. We show that the problem has a unique solution for any non-negative bounded initial value by using Gronwall's inequality. Moreover, the boundedness of the solution is obtained by means of the auxiliary problem. Finally, in the case that the initial data with compactly supported, we analyze the asymptotic behavior of the solution.

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