finite interval $I := (a,b) \subset \mathbb{R}$, and linearly degenerate on $I$, that is, $$\left\{
\begin{array}{ll}
  f''(u) >0 & \; \bigl(u \in (-\infty ,a\, ]\cup [\, b,+\infty )\bigr),\\[5pt]
  f''(u) =0 & \; \bigl(u \in (a,b)\bigr).
\end{array}\right.$$ For the flux function satisfying (1.4), the corresponding Riemann solution does form multiwave pattern which consists of the contact discontinuity with the jump from $u=a$ to $u=b$ and the rarefaction waves, depending on the choice of $a$, $b$, $u_-$ and $u_+$. Thanks to that the cases in which the interval $(a,b)$ is disjoint from the interval $(u_-,u_+)$ are similar as in the case the flux function $f$ is genuinely nonlinear on the whole space $\mathbb{R}$, and the case $u_-<a<u_+<b$ is the same as that for $a<u_-<b<u_+$, we may only consider the typical cases $$a<u_-<b<u_+ \quad \mbox{or} \quad u_-<a<b<u_+.$$ Under the conditions (1.4) and (1.5), they have shown the unique global solution in time to the Cauchy problem (1.3) tends uniformly in space toward the multiwave pattern of the combination of the viscous contact wave and the rarefaction waves as the time goes to infinity. It should be noted that the rarefaction wave which connects the far field states $u_-$ and $u_+$ $\bigl(u_\pm \in (-\infty,a\, 