Existence and stability of traveling wave fronts for a reaction‐diffusion system with spatio‐temporal nonlocal effect

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 97; no. 12; pp. 1555 - 1578
• Main Authors: Wu, Chufen, Li, Mengqi, Weng, Peixuan
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.12.2017
• Subjects: 35B35; 35B40; 35K57; Existence theorems; Fixed points (mathematics); global exponential stability; Mathematical analysis; Perturbation; Reaction‐diffusion system; spatio‐temporal effect; Stability; traveling wave front; Traveling waves; Wave fronts; weighted energy function
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201600170

In this paper, traveling wave fronts for a two dimensional quasi‐monotone reaction‐diffusion system with spatio‐temporal delays are investigated. We use the upper‐lower solution method and the fixed‐point theorem to establish the existence of traveling waves for the system, and the technique of comparison principle combined with the weighted energy function to study the global exponential stability of traveling waves of the equations with large initial perturbation. The initial perturbation around the traveling waves decays exponentially as x→−∞, but it can be arbitrarily large in other locations. The authors use the upper‐lower solution method and the fixed‐point theorem to establish the existence of traveling waves for the system, and the technique of comparison principle combined with the weighted energy function to study the global exponential stability of traveling waves of the equations with large initial perturbation. The initial perturbation around the traveling waves decays exponentially as x →−∞, but it can be arbitrarily large in other locations.