  In this paper, we study the decay rate in time to solutions of the Cauchy
problem for the one-dimensional viscous conservation law where the far field
states are prescribed. Especially, we deal with the case that the flux function
which is convex and also the viscosity is a nonlinearly degenerate one
(p-Laplacian type viscosity). As the corresponding Riemann problem admits a
Riemann solution as the constant state or the single rarefaction wave, it has
already been proved by Matsumura-Nishihara that the solution to the Cauchy
problem tends toward the constant state or the single rarefaction wave as the
time goes to infinity. We investigate that the decay rate in time of the
corresponding solutions. Furthermore, we also investigate that the decay rate
in time of the solution for the higher order derivative. These are the first
result concerning the asymptotic decay of the solutions to the Cauchy problem
of the scalar conservation law with nonlinear viscosity. The proof is given by
L1, L2-energy and time-weighted Lq-energy methods.
