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Asymptotic Behavior of Solutions to the Cauchy Problem for the Scalar Viscous Conservation Law with Partially Linearly Degenerate Flux

In this paper, we investigate the asymptotic behavior in time of solutions to the Cauchy problem for a one-dimensional viscous conservation law where the far field states are prescribed. In particular, we study the case in which the flux function is convex but linearly degenerate on some intervals. When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward the linear combination of the rarefaction waves and viscous contact wave as the time goes to infinity. This is the first result concerning the asymptotics toward multiwave patterns for the Cauchy problem to the scalar viscous conservation law. The proof is given by an $L^2$-energy method and a careful consideration of the interactions between the nonlinear waves. We also show that similar arguments are applicable to the initial-boundary value problem on the half space.