Electron. J. Diff. Equ., Vol. 2013 (2013), No. 179, pp. 1-14.

Exponential stability of traveling fronts for a 2D lattice delayed differential equation with global interaction

Shi-Liang Wu, Tian-Tian Liu

Abstract:
The purpose of this paper is to study traveling wave fronts of a two-dimensional (2D) lattice delayed differential equation with global interaction. Applying the comparison principle combined with the technical weighted-energy method, we prove that any given traveling wave front with large speed is time-asymptotically stable when the initial perturbation around the wave front need decay to zero exponentially as $i \cos\theta +j \sin\theta\to -\infty$, where $\theta$ is the direction of propagation, but it can be allowed relatively large in other locations. The result essentially extends the stability of traveling wave fronts for local delayed lattice differential equations obtained by Cheng et al [1] and Yu and Ruan [16].

Submitted July 1, 2013. Published August 4, 2013.
Math Subject Classifications: 35K57, 35R10, 35B40, 92D25.
Key Words: Exponential stability; traveling wave front; global interaction; lattice differential equation; comparison principle; weighted energy method.

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Shi-Liang Wu
Department of Mathematics, Xidian University
Xi'an, Shaanxi 710071, China
email: slwu@xidian.edu.cn
Tian-Tian Liu
Department of Mathematics, Xidian University
Xi'an, Shaanxi 710071, China
email: ahtian4402@qq.com

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