Sharp relaxation rates for plane waves of general reaction-diffusion models

Abstract

It is a 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 1990s, Goodman [Trans. Am. Math. Soc. 311, 683 (1989)], Kapitula [Trans. Am. Math. Soc. 349, 1901 (1997)], and Xin [Commun. Partial Differ. Equations 17, 1889 (1992)] 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 ≥ 2, algebraic rates are the best possible. In this paper, we revisit this issue. We rigorously calculate the sharp relaxation rates in L∞ based spaces, both for the asymptotic phase and the radiation terms. They 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∞ norms by high order L2 based Sobolev spaces. Finally, we explicitly construct the leading order profiles, both for the phase and for the radiation terms. Our approach relies on the method of scaling variables, as introduced in the work of Gallay and Wayne [Arch. Ration. Mech. Anal. 163, 209 (2002) and Gallay and Wayne, Philos. Trans. R. Soc. A 360, 2155 (2002)] and, in fact, provides sharp relaxation rates in a class of weighted L2 spaces as well.