Front location determines convergence rate to traveling waves

Abstract

We propose a novel method for establishing the convergence rates of solutions to reaction–diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in An et. al. [Arch. Ration. Mech. Anal. 247 (2023), no. 5, article no. 88]. It turns out that the convergence rate is controlled by the distance between the phantom front location for the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the traveling wave itself is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach dramatically simplifies the proof in the Fisher–KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher–KPP case and the exponential rates in the pushed case.