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

TL;DR

This paper investigates the stability of traveling wavefronts in periodic environments, demonstrating exponential stability above a critical speed and algebraic stability at the critical speed, with extensions to time-periodic media and implications for wavefront uniqueness.

Contribution

It introduces a new, accessible method to prove stability of wavefronts in periodic environments, including exponential and algebraic cases, and extends results to time-periodic media.

Findings

Wavefronts are exponentially stable when wave speed exceeds critical value.

Wavefronts are algebraically stable at the critical wave speed.

Stability results imply uniqueness of wavefronts with a given speed.

Abstract

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.