Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations

Rui Huang; Ming Mei; Kaijun Zhang; Qifeng  Zhang

doi:10.3934/dcds.2016.36.1331

Article Contents

2016, Volume 36, Issue 3: 1331-1353. Doi: 10.3934/dcds.2016.36.1331

This issue Previous Article Wandering continua for rational maps Next Article Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations

Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations

1.
School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, 510631
2.
Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2
3.
School of Mathematics and Statistics, Northeast Normal University, Changchun, MO 130024
4.
School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018

Received: December 2014

Revised: March 2015

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.

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

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