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arXiv · 2307.09523

Front location determines convergence rate to traveling waves

Jing An·Christopher Henderson·Lenya Ryzhik
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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 [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.

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Jing An, Christopher Henderson, Lenya Ryzhik. 2023-07-18. Front location determines convergence rate to traveling waves. https://arxiv.org/abs/2307.09523

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