American Institute of Mathematical Sciences

June  2015, 20(4): 1015-1029. doi: 10.3934/dcdsb.2015.20.1015

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

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

Received  April 2014 Revised  October 2014 Published  February 2015

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.
Citation: Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
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