Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 22, pp. 1-24.

Title: Periodic traveling waves and asymptotic spreading of a monostable 
       reaction-diffusion equations with nonlocal effects
       
Authors: Bang-Sheng Han (Southwest Jiaotong Univ., Chengdu, Sichuan China)
         De-Yu Kong     (Southwest Jiaotong Univ., Chengdu, Sichuan China)
         Qihong Shi     (Lanzhou Univ. of Technology, Lanzhou, Gansu, China)
         Fan Wang       (Southwest Jiaotong Univ., Chengdu, Sichuan China)

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.