- f(U^{r}) \, \bigr)  
        + \mu \, \partial_x 
    \left( \, 
    \bigl( \, \partial_x U \, \bigr)^{p} \, 
    \right) \\[5pt]
    \; \; \; \; \; \: \: 
      = \mu \, \partial_x 
      \left( \, 
      \bigl| \, \partial_x (U+U^{r}+\phi) \, \bigr|^{p-1} 
      \partial_x (U+U^{r}+\phi) \,  
      \right) 
  \, \, \: \: \quad  (t>0, x\in \mathbb{R}), \\[10pt]
  \phi(0,x) = \phi_0(x) 
  := u_0(x)-\tilde{U}(0,x)
  \qquad \qquad \; \: \qquad \quad \qquad 
  \; \: \: \; \; \: \; \; \,\, \, \, 
  (x\in \mathbb{R}), 
 \end{array}
 \right.\,\end{aligned}$$ and for any $\epsilon \in (0,1 \,]$, we consider the $\epsilon$-regularized problem as $$\begin{aligned}
 \left\{\begin{array}{ll}
  \partial _t \phi_{\epsilon} + 
  \displaystyle{ 
  \partial_x 
  \Bigl( \, 
  f^{\epsilon} (U^{\epsilon}+U^{r, \epsilon}+\phi_{\epsilon}) 
  - f^{\epsilon}(U^{r, \epsilon}) \, 
  \Bigr) } 
        + \mu \, \partial_x 
    \left( \, 
    \bigl( \, \partial_x U^{\epsilon} \, \bigr)^{p} \, 
    \right)\\[5pt]
    \quad \, \, 
    = \mu \, \partial_x 
      \left( 
      \left( 
      \bigl( \, \partial_x (U^{\epsilon}+U^{r, \epsilon}+\phi_{\epsilon}) \, 
      \bigr)^2 + \epsilon \, 
      \right)^{\frac{p-1}{2}} \partial_x (U^{\epsilon}+U^{r, \epsilon}+\phi_{\epsilon})   
      \right) \\[10pt]
  \: \; \; \; \; \qquad \qquad \qquad \; \: \: \qquad \qquad \qquad \qquad \qquad \qquad 
  \quad \, 
  (t>0, x\in \mathbb{R}), \\[10pt]
  \phi_{\epsilon}(0,x) = \phi_0^{\epsilon}(x) 
  \; \; \; \; \quad \quad \quad \; \: \: \quad \qquad \qquad 
  \qquad \qquad \qquad \quad 
  (x\in \mathbb{R}). 
 \end{array}
 \right.\,\end{aligned}$$ where, $$\begin{aligned}
 \left\{\begin{array}{ll}
\phi^{\epsilon} 