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In this paper, we study the precise decay rate in time to solutions of theCauchy problem for the one-dimensional conservation law with a nonlinearlydegenerate viscosity where the far field states are prescribed. Especially, wedeal with the case when the flux function is convex or concave but linearlydegenerate on some interval. As the corresponding Riemann problem admits aRiemann solution as a multiwave pattern which consists of the rarefaction wavesand the contact discontinuity, it has already been proved by Yoshida that thesolution to the Cauchy problem tends toward the linear combination of therarefaction waves and contact wave for p-Laplacian type viscosity as the timegoes to infinity. We investigate that the decay rate in time of thecorresponding solutions toward the multiwave pattern. Furthermore, we alsoinvestigate that the decay rate in time of the solution for the higher orderderivative. The proof is given by L1, L2-energy and time-weighted Lq-energymethods under the use of the precise asymptotic properties of the interactionsbetween the nonlinear waves.

Decay properties of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with degenerate flux and viscosity