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

Zhao-Xing Yang; Guo-Bao Zhang; Ge Tian; Zhaosheng Feng

doi:10.3934/dcdss.2017029

Article Contents

2017, Volume 10, Issue 3: 581-603. Doi: 10.3934/dcdss.2017029

This issue Previous Article Exponential stability of 1-d wave equation with the boundary time delay based on the interior control Next Article On a hyperbolic-parabolic mixed type equation

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
^* Corresponding author: Guo-Bao Zhang

Received: December 31, 2015

Revised: November 30, 2016

Published: May 2017

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Abstract

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.

Keywords:

Spatially discrete reaction-diffusion equations,
non-monotone traveling waves,
stability,
weighted energy.

Mathematics Subject Classification: Primary: 35K57, 34K20; Secondary: 92D25.

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