EXISTENCE, ASYMPTOTICS AND UNIQUENESS OF TRAVELING WAVES FOR NONLOCAL DIFFUSION SYSTEMS WITH DELAYED NONLOCAL RESPONSE

Abstract

In this paper, we deal with the existence, asymptotic behavior and uniqueness of traveling waves for nonlocal diffusion systems with delay and global response. We first obtain the existence of traveling wave front by using upper-lower solutions method and Schauder's fixed point theorem for $c\gt c_*$ and using a limiting argument for $c=c_*$. Secondly, we find a priori asymptotic behavior of (monotone or non-monotone) traveling waves with the help of Ikehara's Theorem by constructing a Laplace transform representation of a solution. Thirdly, we show that the traveling wave front for each given wave speed is unique up to a translation. Last, we apply our results to two models with delayed nonlocal response.