Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations

Wei-Jie Sheng; Wan-Tong Li

doi:10.3934/dcds.2017115

Article Contents

2017, Volume 37, Issue 5: 2681-2704. Doi: 10.3934/dcds.2017115

This issue Previous Article Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity Next Article Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems

Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations

Wei-Jie Sheng^1, , and
Wan-Tong Li^2,

1.
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Corresponding author
Corresponding author

Received: April 2015

Revised: December 2016

Published: May 2017

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Abstract

This paper is concerned with the stability of time periodic planar traveling fronts of bistable reaction-diffusion equations in multidimensional space. We first show that time periodic planar traveling fronts are asymptotically stable under spatially decaying initial perturbations. In particular, we show that such fronts are algebraically stable when the initial perturbations belong to $L^1$ in a certain sense. Then we further prove that there exists a solution that oscillates permanently between two time periodic planar traveling fronts, which reveals that time periodic planar traveling fronts are not always asymptotically stable under general bounded perturbations. Finally, we address the asymptotic stability of time periodic planar traveling fronts under almost periodic initial perturbations.

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Mathematics Subject Classification: Primary: 35B35, 35K57; Secondary: 35C07, 35B10.

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