Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ

Abstract

This paper is concerned with traveling curved fronts in reaction diffusion equations with degenerate monostable and combustion nonlinearities. For a given admissible pyramidal in three-dimensional spaces, the existence of a pyramidal traveling front has been proved by Wang and Bu [ 30 ] recently. By constructing new supersolutions and developing the arguments of Taniguchi [ 25 ] for the Allen-Cahn equation, in this paper we first characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces, and then establish the uniqueness and asymptotic stability of such three-dimensional pyramidal traveling fronts under the condition that given perturbations decay at infinity.