{\mathopen{{\hbox{$\left}\vbox to 20.5pt{}\right.\n@space$}}}( \left( 
             A-B \left(\frac{y}{t^{\frac{1}{p+1}}} \right)^2 \, 
             \right)\vee 0 {\mathclose{{\hbox{$\left}\vbox to 20.5pt{}\right.\n@space$}}})^{\frac{1}{p-1}}
             \, \mathrm{d}y, 
\end{aligned}\end{aligned}$$ where $$\begin{aligned}
\left\{\begin{array}{ll}
\displaystyle{
A=A_{p,\mu,u_{\pm}}
:= \left( \, 
   \frac{(p-1)\, \left( \, u_{+} - u_{-} \, \right)}
   {8\, \mu\, p(p+1) 
   \biggl( \, \displaystyle{\int^{\frac{\pi}{2}}_{0} 
   \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta }
   \biggr)^2 }
   \, \right)^{\frac{p-1}{p+1}}
}, \\[25pt]
\displaystyle{ 
B=B_{p,\mu}:= \frac{p-1}{2\, \mu \, p(p+1)} 
}, \\[10pt] 
\displaystyle{ 
2A^{\frac{p+1}{2(p-1)}}B^{-\frac{1}{2}}
      \int^{\frac{\pi}{2}}_{0} 
      \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta 
      = u_{+} - u_{-}}
.
\end{array}
\right.\,\end{aligned}$$

Then, we have the next lemma. Because the proofs are very elementary, we omit the proofs.

[**Lemma 2.3.**]{}

Reformulation of the problem
============================

In this section, we reduce our Cauchy problem (1.1) to a simpler case and reformulate the problem in terms of the deviation from the asymptotic state (the same as in [@matsumura-yoshida], [@yoshida]). At first, without loss of generality, we shall consider the case where $a<0$, $b=0$ and the flux function $f(u)$ satisfies $$\left\{
\begin{array}{ll}
  f''(u) >0 & \; (u \in (-\infty ,a]\cup [0,+\infty )),\\[5pt]
  f(u) =0 & \; (u \in (a,0)),
\end{array}\right.$$ under changing the variables and constant as $x-\tilde{\lambda} \,t \mapsto x$, $u-b \mapsto u$, $f(u+b)-f'(b)\,u-f(a) \mapsto f(u)$ and $a-b \mapsto a$ in this order. For the far field states $u_\pm \in \mathbb{R}$, 