Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

Yicheng Jiang; Kaijun Zhang

doi:10.3934/krm.2018048

Article Contents

2018, Volume 11, Issue 5: 1235-1253. Doi: 10.3934/krm.2018048

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Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

Yicheng Jiang and
Kaijun Zhang^,

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

* Corresponding author
* Corresponding author

Received: May 31, 2017

Revised: June 30, 2017

Published: May 2018

The first author is supported by NSFC grant (No.11571066) and the second author is supported by NSFC grant (No.11771071).

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Abstract

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed $c>c_*$, where $c_*>0$ is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves ($c = c_*$), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.

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Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 35K58, 92D25.

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