Propagation phenomenon for bistable parabolic equation in space-periodic environment

This paper is devoted to studying the following spatially periodic reaction–diffusion equation with bistable nonlinearity: ${\partial }_{t}u-\nabla \cdot (A\left(x\right)\nabla u)=f(x,u),t\in R,x\in R^N.$We investigate the effect of spatial heterogeneity on the propagation phenomenon of parabolic equations in $R^N$. As a special entire solution, traveling waves play a significant role in studying the dynamic behavior of reaction–diffusion equations. However, the study of bistable traveling waves in heterogeneous environments is relatively late and sparse. In the present paper, we first establish the uniqueness of wave speed by making use of sub- and super-solution method and comparison principle. Then, based on the uniqueness, the spreading speed is investigated within the framework of the dynamical system.