U \, \right|^{p-1} \partial_x U  \, 
\right)\\
:= 
&-\bigl( \, f'(U+U^r)
- f'(U^r) \, \bigr) \, \partial_x U^r 
- f'(U+U^r)\, \partial_x U \\
&+ \mu \, \partial_x 
\left( \, 
\bigl| \, \partial_x U + \partial_x U^{r} \, \bigr|^{p-1} 
\bigl( \, \partial_x U + \partial_x U^{r} \, \bigr) - 
\left| \, \partial_x U \, \right|^{p-1} \partial_x U  \, 
\right)
\end{aligned}\end{aligned}$$ which consists of the interaction terms of the viscous contact wave $U$ and the approximation of the rarefaction wave $U^r$, and the approximation error of $U^r$ as solution to the conservation law for the $p$-Laplacian type viscosity. Here we should note that $U$ is monotonically nondecreasing and $U^r$ is monotonically increasing, that is, $\partial_x\tilde{U}(t,x) > 0\ \bigl(t\ge 0, x\in \mathbb{R} \bigr)$ which is frequently used hereinafter. Now putting $$u(t,x) = \tilde{U}(t,x) + \phi(t,x)$$ and using (3.5), we can reformulate the problem (1.1) in terms of the deviation $\phi $ from $\tilde{U}$ as $$\begin{aligned}
 \left\{\begin{array}{ll}
  \partial _t\phi + \partial_x \left( f(\tilde{U}+\phi) - f(\tilde{U}) \right) \\[5pt]
  \quad - \mu \, \partial_x 
    \left( \, 
    \bigl| \, \partial_x \tilde{U} + \partial_x \phi \, \bigr|^{p-1} 
    \bigl( \, \partial_x \tilde{U} + \partial_x \phi \, \bigr) - 
    \bigl| \, \partial_x \tilde{U} \, \bigr|^{p-1} \partial_x \tilde{U}  \, 
    \right)\\[2pt]
    \quad 