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

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.