in time by using a very technical energy method and careful estimates of the interactions between the nonlinear waves. Finally, in Section 6, we prove the asymptotic behavior by utilizing the uniform energy estimates in Section 4 and Section 5.

[**Some Notation.**]{} We denote by $C$ generic positive constants unless they need to be distinguished. In particular, use $C(\alpha, \beta, \cdots )$ or $C_{\alpha, \beta, \cdots }$ when we emphasize the dependency on $\alpha, \beta, \cdots $, and $\mathbb{R}^{+}$ as $
\mathbb{R}^{+}:=(0,\infty).$ We also use the Friedrichs mollifier $\rho_\delta \ast $, where, $\rho_\delta(x):=\frac{1}{\delta}\rho \left( \frac{x}{\delta}\right)$ with $$\begin{aligned}
\begin{aligned}
&\rho \in C^{\infty}_0(\mathbb{R}),\quad 
\rho (x)\geq 0\:  (x \in \mathbb{R}), \\
&\mathrm{supp} \{\rho \} \subset 
\left\{x \in \mathbb{R}\: \left|\:  |\, x \, |\le 1 \right. \right\},\quad  
\int ^{\infty}_{-\infty} \rho (x)\, \mathrm{d}x=1, 
\end{aligned}\end{aligned}$$ and $\rho_\delta \ast f$ denote the convolution. For function spaces, $L^p = L^p(\mathbb{R})$ and $H^k = H^k(\mathbb{R})$ denote the usual Lebesgue space and $k$-th order Sobolev space on the whole space $\mathbb{R}$ with norms $||\cdot||_{L^p}$ and $||\cdot||_{H^k}$, respectively. We also define the bounded $C^{m}$-class $\mathscr{B}^{m}$ as follows $$f\in \mathscr{B}^{m}(\Omega)
\, \Longleftrightarrow  \, 
f\in C^m(\Omega), 
\; \sup _{\Omega}\, \sum _{k=0}^{m} \, \bigl| \, D^kf\, \bigr|<\infty$$ for $m< \infty$ and $$f\in \mathscr{B}^{\infty }(\Omega)
