Decay properties of solutions toward a multiwave pattern for the scalar viscous conservation law with partially linearly degenerate flux

In this paper, we study the decay rate in time to solutions toward a multiwave pattern 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 is convex or concave but linearly degenerate on some interval. As the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it has already been proved by Matsumura-Yoshida that the solution to the Cauchy problem tends toward the multiwave pattern which consists of the rarefaction waves and the viscous contact wave as the time goes to infinity. We investigate that the decay rate in time is (1+t)−12(12−1p) in the Lp-norm (2≤p<+∞) and (1+t)−14+ϵ for any ϵ>0 in the L∞-norm if the initial perturbation from the corresponding asymptotics satisfies H1. Furthermore, if the perturbation satisfies H1∩L1, the decay rate in time is (1+t)−12(1−1p)+ϵ for any ϵ>0 in the Lp-norm (1≤p<+∞) and (1+t)−12+ϵ for any ϵ>0 in the L∞-norm. The proof is given by L1, L2-energy and time-weighted Lp-energy methods and the properties of the interactions between the nonlinear waves. We also prove the same results of the decay properties by the similar arguments to the initial-boundary value problem on the half space.