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Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response. (English) Zbl 1311.35053

Authors’ 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 \(c>c_*\) 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.

MSC:

35C07 Traveling wave solutions
35R10 Partial functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
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