# Sharp relaxation rates for plane waves of reaction-diffusion systems

## TL;DR

This paper rigorously determines the optimal algebraic relaxation rates for plane wave solutions in reaction-diffusion systems, improving previous estimates and explicitly constructing the leading profiles using scaling methods.

## Contribution

It provides the first sharp, explicit algebraic relaxation rates in $L^
Infty$ spaces for reaction-diffusion plane waves, surpassing earlier estimates and clarifying the role of dimension.

## Key findings

- Relaxation rates are algebraic and dimension-dependent.
- Sharp relaxation rates are about twice better than previous estimates.
- Explicit leading order profiles for phase and radiation terms are constructed.

## Abstract

It is well-known and classical result that spectrally stable traveling waves of a general reaction-diffusion system in one spatial dimension are asymptotically stable with exponential relaxation rates. In a series of works in the 1990's, the authors have considered plane traveling waves for such systems and they have succeeded in showing asymptotic stability for such objects. Interestingly, the (estimates for the) relaxation rates that they have exhibited, are all algebraic and dimension dependent. It was heuristically argued that as the spectral gap closes in dimensions $n\geq 2$, algebraic rates are the best possible. In this paper, we revisit this issue. We rigorously calculate the sharp relaxation rates in $L^\infty$ based spaces, both for the asymptotic phase and the radiation terms. These turn out to be are indeed algebraic, but about twice better than the best ones obtained in these early works, although this can be mostly attributed to the inefficiencies of using Sobolev embeddings to control $L^\infty$ norms by high order $L^2$ based Sobolev space norms. Finally, we explicitly construct the leading order profiles, both for the phase and the radiation terms. Our approach relies on the method of scaling variables, which provides sharp relaxation rates in a class of weighted $L^2$ spaces as well.