Decay properties of solutions toward shock waves of the scalar conservation law with linear and $p$-Laplacian viscosity

In this paper, we discuss the asymptotic behaviour of weak solutions to the Cauchy problem toward the viscous shock waves for the scalar viscous conservation law. We firstly consider the case that the flux function is the quadratic Burgers flux and obtain the time-decay rate in $L^{\mathrm{\infty }}$-norm for the cases with degenerate $p$-Laplacian viscosity and with linear viscosity, respectively. Moreover, we also give the time-decay rate in $L^{\mathrm{\infty }}$-norm with general flux and linear viscosity. All of these results do not involve smallness conditions on the initial data.