\qquad \qquad \qquad \qquad \qquad \qquad = F_{p}(U,U^r)
  \, \, \: \: \quad  (t>0, x\in \mathbb{R}), \\[5pt]
  \phi(0,x) = \phi_0(x) 
  := u_0(x)-\tilde{U}(0,x) 
  \qquad \qquad \quad \; \: \: \; \; \: \; \; \,\, \, \, (x\in \mathbb{R}). 
 \end{array}
 \right.\,\end{aligned}$$ Then we look for the unique global solution in time $\phi $ which has the asymptotic behavior $$\displaystyle{
\sup_{x \in \mathbb{R}}\left|\,  \phi(t,x) \, \right|
\xrightarrow[\, \, t\to  \infty \, ]
\ \ 0}.$$ Here we note the fact $\phi_0 \in L^2$ and $\partial_x \phi_0 \in L^{p+1}$ by the assumptions on $u_0$ and the fact $$\partial_x \tilde{U}(0,\cdot \, )
=\partial_x U(0,\cdot \, )+\partial_x U^r(0,\cdot \, )\in L^{p+1}.$$ In the following, we always assume that the flux function $f\in C^1(\mathbb{R})\cap C^3 (\, [\, 0,\infty))$ satisfies (3.2), and the far field states satisfy $u_-<0<u_+$. Then the corresponding our main theorem for $\phi$ we should prove is as follows.

[**Theorem 3.2.**]{}

In order to accomplish the proof of Theorem 3.2, we first note that for any $T>0$, the global existence on $[\, 0,T\,]$ and uniqueness can be proved by the similar arguments as in [@matsu-nishi2]. Indeed, we rewrite our Cauchy problem [(3.7)]{} again as $$\begin{aligned}
 \left\{\begin{array}{ll}
    \partial _t \phi + \partial_x \bigl( \, f(U+U^{r}+\phi) 