Existence and Stability of Traveling Waves for Infinite-Dimensional Delayed Lattice Differential Equations

In this paper, we study the existence and stability of traveling waves of infinite-dimensional lattice differential equations with time delay, where the equation may be not quasi-monotone. Firstly, by applying Schauder’s fixed point theorem, we get the existence of traveling waves with the speed c > c∗ (here c∗ is the minimal wave speed). Using a limiting argument, the existence of traveling waves with wave speed c = c∗ is also established. Secondly, for sufficiently small initial perturbations, the asymptotic stability of the traveling waves Φ:={Φ(n+ct)}n∈ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {\Phi }:=\{{\Phi }(n+ct)\}_{n\in \mathbb {Z}}$\end{document} with the wave speed c > c∗ is proved. Here we emphasize that the traveling waves Φ:={Φ(n+ct)}n∈ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {\Phi }:=\{{\Phi }(n+ct)\}_{n\in \mathbb {Z}}$\end{document} may be non-monotone.