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 \(\boldsymbol {\Phi }:=\{{\Phi }(n+ct)\}_{n\in \mathbb {Z}}\) with the wave speed c > c_∗ is proved. Here we emphasize that the traveling waves \(\boldsymbol {\Phi }:=\{{\Phi }(n+ct)\}_{n\in \mathbb {Z}}\) may be non-monotone.

中文翻译：

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无限维时滞格子微分方程行波的存在性和稳定性

本文研究了具有时滞的无限维晶格微分方程行波的存在性和稳定性，其中该方程可能不是准单调的。首先，通过应用Schauder不动点定理，我们得到了速度c > c _∗（这里c _∗是最小波速）的行波的存在。使用一个极限参数，还建立了波速为c = c _∗的行波的存在。其次，对于足够小的初始扰动，行波的渐近稳定性\（\ boldsymbol {\ Phi}：= \ {{{Phi}（n + ct）\} _ {n \ in \ mathbb {Z}} \ ）与波速c> c _∗被证明。在这里，我们强调行波\（\ boldsymbol {\ Phi}：= \ {{\ Phi}（n + ct）\} _ {n \ in \ mathbb {Z}} \\）可能是非单调的。