Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation
Hongmei Cheng, Rong Yuan · Discrete and Continuous Dynamical Systems - B · 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.