Asymptotic Behavior of Solutions Toward a Multiwave Pattern to the Cauchy Problem for the Scalar Conservation Law with the Ostwald--de Waele-Type Viscosity

Abstract

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 the Ostwald--de Waele-type ($p$-Laplacian-type degenerate 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 the Ostwald--de Waele-type viscosity as time goes to infinity. This is the first result concerning the asymptotics toward a multiwave pattern for the Cauchy problem of the scalar conservation law with nonlinear viscosity. The proof is given by a technical energy method and the careful estimates for the interactions between the nonlinear waves.