Time-periodic traveling waves and propagating terraces for multistable equations with a fractional Laplacian: An abstract dynamical systems approach

Abstract

This paper is devoted to studying the existence of traveling wave solutions for reaction-diffusion equations with fractional Laplacians in time-periodic environments. By developing a dynamical systems approach, we establish the existence of propagating terraces for equations with multistable nonlinearities, which consequently implies the existence of bistable traveling waves. Furthermore, we investigate whether traveling waves exist in the multistable case. We provide an affirmative answer to this question for a special type of nonlinearity in high-frequency oscillating environments, and determine the homogenized limit of the traveling waves as the period approaches 0.