Periodic pyramidal traveling fronts of bistable reaction–diffusion equations with time-periodic nonlinearity

This paper deals with the existence and stability of periodic pyramidal traveling fronts for reaction–diffusion equations with bistable time-periodic nonlinearity in RN with N⩾3. It is well known that two-dimensional periodic traveling curved fronts exist and are stable. In this paper, by constructing various of supersolutions and subsolutions, we first show that there exist three-dimensional periodic pyramidal traveling fronts, and then we prove that such periodic pyramidal traveling fronts are asymptotically stable. Finally, we further prove that our existence result holds for RN with N⩾4.