# American Institute of Mathematical Sciences

June  2017, 10(3): 581-603. doi: 10.3934/dcdss.2017029

## Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay

 1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China 2 School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA

* Corresponding author: Guo-Bao Zhang

Received  January 2016 Revised  December 2016 Published  February 2017

This paper is concerned with traveling waves for temporally delayed, spatially discrete reaction-diffusion equations without quasi-monotonicity. We first establish the existence of non-critical traveling waves (waves with speeds c>c*, where c* is minimal speed). Then by using the weighted energy method with a suitably selected weight function, we prove that all noncritical traveling waves Φ(x+ct) (monotone or nonmonotone) are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small.

Citation: Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
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