  In this paper, we study the precise decay rate in time to solutions of the
Cauchy problem for the one-dimensional conservation law with a nonlinearly
degenerate viscosity where the far field states are prescribed. Especially, we
deal with the case when the flux function is convex or concave but linearly
degenerate on some interval. As the corresponding Riemann problem admits a
Riemann solution as a multiwave pattern which consists of the rarefaction waves
and the contact discontinuity, it has already been proved by Yoshida that the
solution to the Cauchy problem tends toward the linear combination of the
rarefaction waves and contact wave for p-Laplacian type viscosity as the time
goes to infinity. We investigate that the decay rate in time of the
corresponding solutions toward the multiwave pattern. Furthermore, we also
investigate that the decay rate in time of the solution for the higher order
derivative. The proof is given by L1, L2-energy and time-weighted Lq-energy
methods under the use of the precise asymptotic properties of the interactions
between the nonlinear waves.
