# On Classical Solutions for A Kuramoto-Sinelshchikov-Velarde-Type Equation
## 2 Proof of Theorem 1, under the Assumptions (2) and (4)
\begin{cases}\partial_{t}\omega+2\gamma\left[(\partial_{x}u_{1})^{2}-( \partial_{x}u_{2})^{2}\right]+\delta\partial_{x}^{3}\omega+\mu\partial_{x}^{ 2}\omega\\ \qquad\qquad\qquad+\beta^{2}\partial_{x}^{4}\omega+\gamma u_{1}\partial_{x}^{ 2}u_{1}-\gamma u_{2}\partial_{x}^{2}u_{2}=0,&t>0,\,x\in\mathbb{R},\\ \omega(0,x)=u_{1,0}(x)-u_{2,0}(x),&x\in\mathbb{R}.\end{cases}  \tag{55}\]  
Observe that thanks to (36),  
\[u_{1}\partial_{x}^{2}u_{1}-u_{2}\partial_{x}^{2}u_{2}=u_{1} \partial_{x}^{2}u_{1}-u_{2}\partial_{x}^{2}u_{1}+u_{2}\partial_{x}^{2}u_{1}-u_ {2}\partial_{x}^{2}u_{2}=\omega\partial_{x}^{2}u_{1}+u_{2}\partial_{x}^{2}\omega,\] \[(\partial_{x}u_{1})^{2}-(\partial_{x}u_{2})^{2}=(\partial_{x}u_{1} +\partial_{x}u_{2})\partial_{x}\omega.\]Consequently, (55) is equivalent to the following equation:  
\[\partial_{t}\omega+2\gamma(\partial_{x}u_{1}+\partial_{x}u_{2})\partial_{x} \omega+\gamma\omega\partial_{x}^{2}u_{1}+\gamma u_{2}\partial_{x}^{2}\omega+ \delta\partial_{x}^{3}\omega+\mu\partial_{x}^{2}\omega+\beta^{2}\partial_{x}^{ 4}\omega=0. \tag{56}\]  
Therefore, multiplying (56), by (39) and an integration on \(\mathbb{R}\) of, we have that  
\[ \begin{split}\frac{d}{dt}\left\|\omega(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}&+2\beta^{2}\left\|\partial_{x}^{2}\omega(t, \cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\\ =& 2\mu\left\|\partial_{x}\omega(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}-2\gamma\int_{\mathbb{R}}(\partial_{x}u_{1}+\partial_{x}u_{2 })\omega\partial_{x}\omega dx\\ &-2\gamma\int_{\mathbb{R}}\partial_{x}^{2}u_{1}\omega^{2}dx-2 \gamma\int_{\mathbb{R}}u_{2}\omega\partial_{x}^{2}\omega dx.\end{split}  \tag{57}\]  
Since \(u_{1}\), \(u_{2}\in H^{3}(\mathbb{R})\), for every \(0\leq t\leq T\), we have  
\[ \begin{split}\left\|\partial_{x}u_{1}\right\|_{L^{\infty}((0,T) \times\mathbb{R})},\,\left\|\partial_{x}u_{2}\right\|_{L^{\infty}((0,T)\times \mathbb{R})}&\leq C(T),\\ \left\|\partial_{x}^{2}u_{1}\right\|_{L^{\infty}((0,T)\times \mathbb{R})},\,\left\|u_{2}\right\|_{L^{\infty}((0,T)\times\mathbb{R})}& \leq C(T).\end{split}  \tag{58}\]  
Due to (58) and the Young inequality,  
\[ \begin{split} 2|\gamma|&\int_{\mathbb{R}}(\partial_{x}u_{ 1}+\partial_{x}u_{2})\omega\partial_{x}\omega dx\\ &\leq\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \gamma^{2}\int_{\mathbb{R}}(\partial_{x}u_{1}+\partial_{x}u_{2})^{2}(\partial _{x}\omega)^{2}dx\\ &\leq\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \gamma^{2}\left(\left\|\partial_{x}u_{1}\right\|_{L^{\infty}((0,T)\times \mathbb{R})}+\left\|\partial_{x}u_{2}\right\|_{L^{\infty}((0,T)\times\mathbb{ R})}\right)\left\|\partial_{x}\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\\ &\leq\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+C(T) \left\|\partial_{x}\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2},\\ 2|\gamma|&\int_{\mathbb{R}}|\partial_{x}^{2}u_{1}| \omega^{2}dx\\ &\leq 2|\gamma|\left\|\partial_{x}^{2}u_{1}\right\|_{L^{\infty}((0,T) \times\mathbb{R})}\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq C (T)\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2},\\ 2|\gamma|&\int_{\mathbb{R}}|u_{2}\omega||\partial_{ x}^{2}\omega|dx\\ &=2\int_{\mathbb{R}}\left|\frac{\gamma u_{2}\omega}{\beta^{2}} \right|\left|\beta\partial_{x}^{2}\omega\right|dx\\ &\leq\frac{\gamma^{2}}{\beta^{2}}\left\|u_{2}\right\|_{L^{\infty} ((0,T)\times\mathbb{R})}\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2} +\beta^{2}\left\|\partial_{x}^{2}\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})} ^{2}\\ &\leq C(T)\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+ \beta^{2}\left\|\partial_{x}^{2}\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2 }.\end{split} \]  
It follows from (57) that  
\[ \begin{split}\frac{d}{dt}\left\|\omega(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}+\beta^{2}\left\|\partial_{x}^{2}\omega(t,\cdot)\right\|_{L^{2 }(\mathbb{R})}^{2}\\ \leq& C(T)\left\|\omega(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}+C(T)\left\|\partial_{x}\omega(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}.\end{split} \]  
Therefore, arguing as in Section 2, we have (16).