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

In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with the Ostwald-de Waele type viscosity (p-Laplacian type degenerate viscosity) ∂tu+∂x(f(u))=μ∂x(|∂xu|p−1∂xu) 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. When the corresponding Riemann problem admits 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 the Ostwald-de Waele type viscosity as time goes to infinity. We investigate the decay rate in time of the solution toward the multiwave pattern. Furthermore, we investigate the decay rate in time of the solution for the 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.