Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation

Hongmei Cheng; Rong Yuan

doi:10.3934/dcdsb.2015.20.1015

Article Contents

2015, Volume 20, Issue 4: 1015-1029. Doi: 10.3934/dcdsb.2015.20.1015

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Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation

Hongmei Cheng^1, and
Rong Yuan^2,

1.
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875
2.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received: March 31, 2014

Revised: September 30, 2014

Published: May 2015

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Abstract

This paper is concerned with the asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equation on $\mathbb{R}^n$, $n\geq 4$. Our first result states that pyramidal traveling fronts are asymptotically stable under the initial perturbations that decay at space infinity. Then we further show the existence of a solution that oscillates permanently between two pyramidal traveling fronts, which implies that pyramidal traveling fronts are not asymptotically stable under more general perturbations. Our main technique is the supersolution and subsolution method coupled with the comparison principle.

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Mathematics Subject Classification: 35K57, 35B10, 35B35, 35C07.

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