Average speeds of time almost periodic traveling waves for rapidly/slowly oscillating reaction-diffusion equations

TL;DR

This paper investigates the asymptotic behavior of average wave speeds in time almost periodic reaction-diffusion equations, revealing how rapid or slow oscillations influence wave propagation and convergence to homogenized or averaged speeds.

Contribution

It provides new insights into the asymptotic average speeds of traveling waves in almost periodic environments, including explicit formulas and convergence rate estimates.

Findings

In rapidly oscillating environments, average speeds converge to the homogenized wave speed.

In slowly oscillating environments, average speeds approximate the arithmetic mean of frozen coefficient speeds.

Temporal variations significantly influence wave propagation and can alter directions even in periodic settings.

Abstract

This paper is concerned with the propagation dynamics of time almost periodic reaction-diffusion equations. Assuming the existence of a time almost periodic traveling wave connecting two stable steady states, we focus especially on the asymptotic behavior of average wave speeds in both rapidly oscillating and slowly oscillating environments. We prove that, in the rapidly oscillating case, the average speed converges to the constant wave speed of the homogenized equation; while in the slowly oscillating case, it approximates the arithmetic mean of the constant wave speeds for a family of equations with frozen coefficients. In both cases, we provide estimates on the convergence rates showing that, in comparison to the limiting speeds, the deviations of average speeds for almost periodic traveling waves are at most linear in certain sense. Furthermore, our explicit formulas for the limiting speeds indicate that temporal variations have significant influences on wave propagation. Even in periodic environments, it can alter the propagation direction of bistable equations.