Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 84, pp. 1-23.

Title: Spatial dynamics of a nonlocal bistable reaction diffusion equation
       
Authors: Bang-Sheng Han (Southwest Jiaotong Univ., Chengdu, Sichuan, China)
         Meng-Xue Chang (Southwest Jiaotong Univ., Chengdu, Sichuan, China)
         Yinghui Yang   (Southwest Jiaotong Univ., Chengdu, Sichuan, China)

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.