In this talk, we consider the initial value problem for a one-dimensional viscous conservation law. It is well known that the solution of the corresponding Riemann problem is closely related to the asymptotic behavior of solutions, and that its structure strongly depends on the properties of the nonlinear flux function. In the case where the solution of the Riemann problem consists of an Oleinik shock and a rarefaction wave, we establish the asymptotic stability of the composite wave composed of the viscous Oleinik shock and the rarefaction wave. The proof is based on the energy method, and the desired a priori estimate is obtained by introducing a time-dependent shift to the nonlinear waves. The results presented in this talk are based on joint work with Yoshihiro Ueda (Kobe University).