_{0}(x) := ( \rho_{\epsilon} \underset{x}{\ast } \phi_{0} ) (x) 
                        := \displaystyle{\int_{-\infty}^{\infty} 
                           \rho_{\epsilon} (x-y) \, 
                           \phi_{0}(y) \, \mathrm{d}y} 
                           \in H^{\infty}(\mathbb{R}_{x}), \\[10pt]
f^{\epsilon} (u) := ( \rho_{\epsilon} \underset{u}{\ast } f ) (u) 
                 := \displaystyle{\int_{-\infty}^{\infty} 
                    \rho_{\epsilon} (u-v) \, 
                    f(v) \, \mathrm{d}v}
                    \in C^{\infty}(\mathbb{R}_{u}), \\[10pt]
U^{r, \epsilon} (t,x) := \left( \bigl( f^{\epsilon} \bigr)' \right)^{-1} 
                         \Bigl( 
                         w\bigl(\, t,x\: 
                         ;\: f^{\epsilon}(0), \, f^{\epsilon}(w_+) \, \bigr) 
                         \Bigr)
                      \in \mathscr{B}^{\infty}
                          \bigl(\, [ \, 0, \infty)_{t} 
                          \times \mathbb{R}_{x} \bigr), \\[10pt]
U^{\epsilon} (t,x) := ( \rho_{\epsilon} \underset{x}{\ast } U ) (t,x) 
                   := \displaystyle{\int_{-\infty}^{\infty} 
                      \rho_{\epsilon} (x-y) \, 
                                            U\left( \frac{y}{(1+t)^{\frac{1}{p+1}}} \right)
                      \, \mathrm{d}y} \\[12pt]
                      \qquad \qquad \qquad \qquad \quad \quad \quad 
                      \in \mathscr{B}^{1} \cap C^{\infty}
                          \bigl(\, (-1, \infty)_{t} \times \mathbb{R}_{x} \bigr).
\end{array}
 \right.\,\end{aligned}$$ Here, $w(t,x)$ is the classical solution of (2.4) with $w_- = 0$. If we define $u_{\epsilon} := \phi_{\epsilon} + U^{\epsilon} +U^{r, \epsilon}$, we also have the equivalent form $$\begin{aligned}
 \left\{\begin{array}{ll}
  \partial _t u_{\epsilon} + 
  \displaystyle{ 
  \partial_x 
  \Bigl( \, 
  f^{\epsilon} (u_{\epsilon}) \, 
  \Bigr) } 
    = \mu \, \partial_x 
      \left( 
      \left( 
      \bigl( \, \partial_x u_{\epsilon} \, 
      \bigr)^2 + \epsilon \, 
      \right)^{\frac{p-1}{2}} \partial_x u_{\epsilon} \, 
      \right) \\[10pt]
  \, \qquad \qquad \qquad 
  \qquad \qquad \qquad \qquad \qquad \qquad 
  \quad \, 
  (t>0, x\in \mathbb{R}), \\[10pt]
  u_{\epsilon}(0,x) = u_0^{\epsilon}(x) 
                    := \phi_0^{\epsilon}(x) 
                       + U^{\epsilon}(0,x)
                       + U^{r, 