Front location determines convergence rate to traveling waves

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 [2]. 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 itself has a pulled nature, regardless of whether the traveling wave is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach simplifies dramatically 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.