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

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. For more information see https://ejde.math.txstate.edu/Volumes/2021/22/abstr.html