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

This paper deals with the multidimensional stability of time periodic V-shaped traveling fronts in bistable reaction diffusion equations. It is well known that time periodic V-shaped traveling fronts are asymptotically stable in two dimensional space. In the current study, we further show that such fronts are asymptotically stable under spatially decaying initial perturbations in R-n with n >= 3. In particular, we show that the fronts are algebraically stable if the initial perturbations belong to L-1 in a certain sense. Furthermore, we prove that there exists a solution oscillating permanently between two time periodic V-shaped traveling fronts, which implies that time periodic V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Finally we show that time periodic V-shaped traveling fronts are only time global solutions of the Cauchy problem if the initial perturbations lie between two time periodic V-shaped traveling fronts. (C) 2016 Elsevier Ltd. All rights reserved.