]$ or $u_\pm \in [\, b,\infty)\bigr)$ is explicitly given by $$u=u^r \left( \frac{x}{t}\: ;\:  u_- , u_+ \right)
:= \left\{
\begin{array}{ll}
  u_-  & \; \bigl(\, x \leq \lambda(u_-)\,t \, \bigr),\\[7pt]
  \displaystyle{ (\lambda)^{-1}\left( \frac{x}{t}\right) } 
  & \; \bigl(\, \lambda(u_-)\,t \leq x \leq \lambda(u_+)\,t\,  \bigr),\\[7pt]
   u_+ & \; \bigl(\, x \geq \lambda(u_+)\,t \, \bigr),
\end{array}
\right.$$ where $\lambda(u):=f'(u)$, and the viscous contact wave which connects $u_-$ and $u_+$ ($u_\pm \in [\,a,b\,]$) is given by an exact solution of the linear convective heat equation $$  \partial _t u + \tilde{\lambda} \, \partial_x u = \mu \, \partial_x^2 u
  \qquad \biggl(
  \,\tilde{\lambda}:= \frac{f(b)-f(a)}{b-a},\ 
  t>0,x \in  \mathbb{R} 
  \biggr)
$$ which has the form $$u=U\left(\frac{x-\tilde{\lambda} \, t}{\sqrt{t}}\: ;\: u_- ,u_+ \right)$$ where $U\left(\frac{x}{\sqrt{t}}\,;\,u_- ,u_+ \right)$ is explicitly defined by $$U\left(\frac{x}{\sqrt{t}}\: ;\: u_- ,u_+ \right)
 :=u_- +\frac{u_+ - u_-}{\sqrt{\pi}}
   {\mathlarger{\int}}^{\frac{\mathlarger{x}}{\mathlarger{\sqrt{4\mu t}}}}_{-\infty} 
 \mathrm{e}^{-\xi^2}\, \mathrm{d}\xi 
\quad \, (t>0, x\in \mathbb{R}).$$ Yoshida [@yoshida] also obtained the precise decay properties for the asymptotics (cf. [@hashimoto-kawashima-ueda]). In the proof of them, the a priori energy estimates acquired by an $L^2$-energy method and careful estimates for the terms of nonlinear interactions of the viscous contact wave and the rarefaction waves. The aim of the present paper is to extend the results in the 