(1.3) tends toward the rarefaction wave; if $f''(u) < 0\ (u\in \mathbb{R})$, that is, the Riemann solution consists of a single shock wave solution, the global solution of the Cauchy problem (1.3) does the corresponding smooth traveling wave solution (“viscous shock wave”) of (1.3) with a spacial shift (cf. [@ilin-kalashnikov-oleinik]). More generally, in the case of the flux functions which are not uniformly genuinely nonlinear, when the Riemann solution consists of a single shock wave satisfying Ole[ĭ]{}nik’s shock condition, Matsumura-Nishihara [@matsu-nishi3] showed the asymptotic stability of the corresponding viscous shock wave. However, when we consider the circumstances where the Riemann solution generically forms a pattern of multiple nonlinear waves which consists of rarefaction waves, shock waves and waves of contact discontinuity (refer to [@liep-rosh]), there had been no results about the asymptotics toward the multiwave pattern. Recently, Matsumura-Yoshida [@matsumura-yoshida] proved the asymptotics toward a multiwave pattern of the superposition of the rarefaction waves and a self-similar solution (“viscous contact wave”) which is corresponded to the wave of the contact discontinuity. Namely, they investigated the case where the flux function $f$ is smooth and genuinely nonlinear (that is, $f$ is convex function or concave function) on the whole $\mathbb{R}$ except a 