  In this paper, we investigate the asymptotic behavior of solutions toward a
multiwave pattern of the Cauchy problem for the scalar viscous conservation law
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, and also the viscosity is a nonlinearly degenerate one (p-Laplacian
type viscosity). 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 contact wave for p-Laplacian type
viscosity as the time goes to infinity. This is the first result concerning the
asymptotics toward multiwave pattern for the Cauchy problem of the scalar
conservation law with nonlinear viscosity. The proof is given by a technical
energy methods and the careful estimates for the interactions between the
nonlinear waves.
