$u_-<0<u_+$. The corresponding main theorem is the following.

[**Theorem 3.1.**]{}

Here, we first should note by Lemma 2.2 and Lemma 2.3, the asymptotic state $U_{multi}(\, t,x\: ; \: u_-, u_+)$ can be replaced by a following approximated one $$\tilde{U}(t,x) := U(1+t,x) + U^r(t,x)$$ where $$U(1+t,x)=U\left( \frac{x}{(1+t)^{\frac{1}{p+1}}}\: ;\: u_- ,0 \right),
\quad U^r(t,x) =U^r(\, t,x\: ; \: 0, u_+).$$ This is because, from Lemma 2.2 (especially (4)) and Lemma 2.3 (especially (vi)), $$\begin{aligned}
\begin{aligned}
&\sup_{x\in \mathbb{R}}\, \left| \, \tilde{U}(t,x) - U_{multi}(t,x) \, \right| 
\le
\sup_{x\in \mathbb{R}} \, \bigl| \, U(1+t,x) - U(t,x) \, \bigr|\\ 
&\qquad \qquad + \sup_{x\in \mathbb{R}}\, 
\left| \, U^r(t,x) - u^r\left( \frac{x}{t}\right) \, \right|
\to 0\quad (t \to \infty).
\end{aligned}\end{aligned}$$ In the following, we write $U(1+t,x)$ again $U(t,x)$ for simplicity. Then it is noted that $\tilde{U}$ approximately satisfies the equation of (1.1) as $$\partial _t\tilde{U} +\partial_x \bigl(f(\tilde{U})\bigr)
 - \mu \, 
 \partial_x 
\left( \, 
\bigl| \, \partial_x \tilde{U} \, \bigr|^{p-1} \partial_x \tilde{U} \, 
\right)
 = -F_{p}(U,U^r),$$ where the remainder term $F_{p}(U,U^r)$ is explicitly given by $$\begin{aligned}
\begin{aligned}
F_{p}(U,U^r)
:= 
&\widetilde{F_{p} }(U,U^r)\\
&+ \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 