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

This paper is concerned with the stability of time periodic pla- nar traveling fronts of bistable reaction-diffusion equations in multidimensional space. We first show that time periodic planar traveling fronts are asymptot- ically stable under spatially decaying initial perturbations. In particular, we show that such fronts are algebraically stable when the initial perturbations belong to L1 in a certain sense. Then we further prove that there exists a so- lution 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 ad- dress the asymptotic stability of time periodic planar traveling fronts under almost periodic initial perturbations.