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In this paper, we study the decay rate in time to solutions of the Cauchyproblem for the one-dimensional viscous conservation law where the far fieldstates are prescribed. Especially, we deal with the case that the flux functionwhich is convex and also the viscosity is a nonlinearly degenerate one(p-Laplacian type viscosity). As the corresponding Riemann problem admits aRiemann solution as the constant state or the single rarefaction wave, it hasalready been proved by Matsumura-Nishihara that the solution to the Cauchyproblem tends toward the constant state or the single rarefaction wave as thetime goes to infinity. We investigate that the decay rate in time of thecorresponding solutions. Furthermore, we also investigate that the decay ratein time of the solution for the higher order derivative. These are the firstresult concerning the asymptotic decay of the solutions to the Cauchy problemof the scalar conservation law with nonlinear viscosity. The proof is given byL1, L2-energy and time-weighted Lq-energy methods.

Decay properties of solutions to the Cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity