# On Classical Solutions for A Kuramoto-Sinelshchikov-Velarde-Type Equation
## 2 Proof of Theorem 1, under the Assumptions (2) and (4)
\begin{split} 2|\gamma|\int_{\mathbb{R}}(\partial_{x}u)^{2}| \partial_{t}u|dx=& 2\int_{\mathbb{R}}\left|\frac{\gamma(\partial_{x}u)^{2}}{ \sqrt{D_{3}}}\right|\left|\sqrt{D_{3}}\partial_{t}u\right|dx\\ \leq&\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t, \cdot)\right\|_{L^{4}(\mathbb{R})}^{4}+D_{3}\left\|\partial_{t}u(t,\cdot) \right\|_{L^{2}(\mathbb{R})}^{2},\\ 2|\delta|\int_{\mathbb{R}}|\partial_{x}^{3}u||\partial_{t}u|dx=& 2\int_{\mathbb{R}}\left|\frac{ \partial_{x}^{3}u}{\sqrt{D_{3}}}\right|\left|\sqrt{D_{3}}\partial_{t}u\right| dx\\ \leq&\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{3}u (t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+D_{3}\left\|\partial_{t}u(t,\cdot) \right\|_{L^{2}(\mathbb{R})}^{2},\\ 2|\mu|\int_{\mathbb{R}}|\partial_{x}^{2}u||\partial_{t}u|dx=& 2\int_{\mathbb{R}}\left|\frac{ \mu\partial_{x}^{2}u}{\sqrt{D_{3}}}\right|\left|\sqrt{D_{3}}\partial_{t}u \right|dx\\ \leq&\frac{\mu^{2}}{D_{3}}\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+D_{3}\left\|\partial_{t}u(t,\cdot) \right\|_{L^{2}(\mathbb{R})}^{2},\\ 2|\gamma|\int_{\mathbb{R}}|u\partial_{x}^{2}u||\partial_{t}u|dx=& 2\int_{ \mathbb{R}}\left|\frac{\gamma u\partial_{x}^{2}u}{\sqrt{D_{3}}}\right|\left| \sqrt{D_{3}}\partial_{t}u\right|dx\\ \leq&\frac{\gamma^{2}}{D_{3}}\int_{\mathbb{R}}u^{2}( \partial_{x}^{2}u)^{2}dx+D_{3}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}\\ \leq&\frac{\gamma^{2}}{D_{3}}\left\|u\right\|_{L^{ \infty}((0,T)\times\mathbb{R})}^{2}\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{ L^{2}(\mathbb{R})}^{2}+D_{3}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}\\ \leq&\frac{C(T)}{D_{3}}\left\|\partial_{x}^{2}u(t, \cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+D_{3}\left\|\partial_{t}u(t,\cdot) \right\|_{L^{2}(\mathbb{R})}^{2},\end{split} \]where \(D_{3}\) is a positive constant, which will be specified later. Consequently, by (50),  
\[\beta^{2}\frac{d}{dt}\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{ 2}(\mathbb{R})}^{2}+2\left(1-2D_{3}\right)\left\|\partial_{t}u(t,\cdot)\right\| _{L^{2}(\mathbb{R})}\] \[\qquad\leq\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{4}(\mathbb{R})}^{4}+\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{ 3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{C(T)}{D_{3}}\left\|\partial _{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]  
Taking \(D_{3}=\frac{1}{2}\), we have  
\[\beta^{2}\frac{d}{dt}\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L ^{2}(\mathbb{R})}^{2}+\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}\] \[\qquad\leq 2\gamma^{2}\left\|\partial_{x}u(t,\cdot)\right\|_{L^{4} (\mathbb{R})}^{4}+2\delta^{2}\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2} (\mathbb{R})}^{2}+C(T)\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}.\]  
It follows from (4), (45), (47) that  
\[\beta^{2}\left\|\partial_{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{ R})}^{2}+\int_{0}^{t}\left\|\partial_{t}u(s,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}ds\] \[\qquad\leq\left\|u_{0}\right\|_{H^{2}(\mathbb{R})}^{2}+2\gamma^{2 }\int_{0}^{t}\left\|\partial_{x}u(s,\cdot)\right\|_{L^{4}(\mathbb{R})}^{4}ds\] \[\qquad+2\delta^{2}\int_{0}^{t}\left\|\partial_{x}^{3}u(s,\cdot) \right\|_{L^{2}(\mathbb{R})}^{2}ds+C(T)\int_{0}^{t}\left\|\partial_{x}^{2}u(s,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}ds\leq C(T),\]  
which gives (49). **Lemma 9**: _Fix \(T>0\) and assume (5). There exists a positive constant \(C(T)>0\), such that_  
\[\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\beta^{2} \int_{\mathbb{R}}\left\|\partial_{x}^{5}u(s,\cdot)\right\|_{L^{2}(\mathbb{R}) }^{2}ds\leq C(T), \tag{51}\]  
_for every \(0\leq t\leq T\). In particular, we have that_  
\[\left\|\partial_{x}^{2}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq C(T). \tag{52}\]  
Let \(0\leq t\leq T\).