Electron. J. Differential Equations, Vol. 2021 (2021), No. 22, pp. 1-24.

Periodic traveling waves and asymptotic spreading of a monostable reaction-diffusion equations with nonlocal effects

Bang-Sheng Han, De-Yu Kong, Qihong Shi, Fan Wang

Abstract:
This article concerns the dynamical behavior for a reaction-diffusion equation with integral term. First, by using bifurcation analysis and center manifold theorem, the existence of periodic steady-state solution are established for a special kernel function and a general kernel function respectively. Then, we prove the model admits periodic traveling wave solutions connecting this periodic steady state to the uniform steady state u=1 by applying center manifold reduction and the analysis to phase diagram. By numerical simulations, we also show the change of the wave profile as the coefficient of aggregate term increases. Also, by introducing a truncation function, a shift function and some auxiliary functions, the asymptotic behavior for the Cauchy problem with initial function having compact support is investigated.

Submitted August 27, 2020. Published March 30, 2021.
Math Subject Classifications: 35C07, 35B10, 35B40, 35R09, 92D25.
Key Words: Reaction-diffusion; nonlocal delay; periodic traveling wave; asymptotic behavior; numerical simulation, critical exponent.
DOI: https://doi.org/10.58997/ejde.2021.22

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Bang-Sheng Han
School of Mathematics
Southwest Jiaotong University
Chengdu, Sichuan 611756, China
email: hanbangsheng@swjtu.edu.cn
De-Yu Kong
School of Mathematics
Southwest Jiaotong University
Chengdu, Sichuan 611756, China
email: KongDY@my.swjtu.edu.cn
Qihong Shi
Department of Mathematics
Lanzhou University of Technology
Lanzhou, Gansu 730000, China
email: shiqh@lut.edu.cn
Fan Wang
School of Mathematics
Southwest Jiaotong University
Chengdu, Sichuan 611756, China
email: wangf767@swjtu.edu.cn

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