we only deal with the typical case $a<u_-<0<u_+$ for simplicity, since the case $u_-<a<0<u_+$ can be treated technically in the same way of the proof as $a<u_-<0<u_+$. Indeed, in the case $u_-<a<0<u_+$, as we shall see in Section 4 and Section 5, there appears the extra nonlinear interaction terms between two rarefaction waves $u^r(\, \frac{x}{t}\: ;\: u_-,a)$ and $u^r(\, \frac{x}{t}\: ;\: 0,u_+)$ with $\lambda(a)=\lambda(0)=0$ in the remainder term of the viscous conservation law for the asymptotics $U_{multi}$(see the right-hand side of (3.5)). These terms can be handled in much easier way by Lemma 2.2 than that for other essential nonlinear interaction terms between the rarefaction and the viscous contact waves. Furthermore, we should point out that the problem under the assumptions for the flux function (3.1) and the far field states $a<u_-<0<u_+$ is essentially the same as that for $a=-\infty$, because obtaining the a priori and the uniform energy estimates for the former one can be given in almost the same way as the latter one. Therefore, it is quite natural for us to treat only a simple case $$\left\{
\begin{array}{ll}
  f''(u) >0 & \; (u \in [\, 0,\infty )),\\[5pt]
  f(u) =0 & \; (u \in (-\infty,0)),
\end{array}\right.$$ and assume 