Exponential and Algebraical Stability of Traveling Wavefronts in Periodic Spatial-Temporal Environments

Global stability of traveling wavefronts in a periodic spatial-temporal environment in $n$-dimension ($n\ge 1$) is studied. The wavefront is proved to be exponentially stable in the form of $ O(e^{-\mu t})$ for some $\mu>0$, when the wave speed is greater than the critical one, and algebraically stable in the form of $O(t^{-n/2})$ in the critical case. A new and easy to follow method is developed. These results are then extended to the case of time-periodic media. Finally, we illustrate how the stability result can be directly used to obtain the uniqueness of the wavefront with a given speed.