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

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 (pp-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 and their derivative. These are the first results concerning the asymptotic decay of the solutions and their derivative to the Cauchy problem of the scalar conservation law with nonlinear viscosity. The proof is given by L1L1, L2L2-energy and time-weighted LqLq-energy methods.