Periodic traveling waves and propagating terraces for multistable nonlinearity in cylinders

This article is devoted to the study of a class of reaction-advection-diffusion equations of the form u_t−Δu+β(y)u_x=f(t,u) in cylinders, where f is a tristable or multistable nonlinearity satisfying f(⋅,0)=f(⋅,1)=0. We establish the alternative of wave solutions, that is, either there is a unique, asymptotically stable periodic traveling wave connecting 0 to 1 directly, or there is a unique propagating terrace connecting 0 to 1. Under the assumption that all wave speeds in propagating terrace are not equal to each other, we further obtain that the propagating terrace is asymptotically stable if it exists. Moreover, the sufficient and necessary conditions for the existence of periodic traveling waves connecting 0 to 1 are given respectively. Our arguments are based on the parabolic maximum principle, sliding method and the super- and sub-solution method.