`a`
Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

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

Pages: 581 - 603, Volume 10, Issue 3, June 2017      doi:10.3934/dcdss.2017029

 
       Abstract        References        Full Text (453.9K)       Related Articles       

Zhao-Xing Yang - College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China (email)
Guo-Bao Zhang - School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China (email)
Ge Tian - College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China (email)
Zhaosheng Feng - School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, United States (email)

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 $\phi(x+ct)$ (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.

Received: January 2016;      Revised: December 2016;      Available Online: February 2017.

 References