  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.
