# On Classical Solutions for A Kuramoto-Sinelshchikov-Velarde-Type Equation
## 2 Proof of Theorem 1, under the Assumptions (2) and (4)
Multiplying (44) by \(-2\partial_{x}^{6}u\), an integration on \(\mathbb{R}\) gives  
\[\frac{d}{dt}\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2} =-2\int_{\mathbb{R}}\partial_{x}^{6}u\partial_{t}udx\] \[\qquad\leq\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{4}(\mathbb{R})}^{4}+\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{ 3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{C(T)}{D_{3}}\left\|\partial _{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]0 \[\qquad\leq\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{4}(\mathbb{R})}^{4}+\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{ 3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{C(T)}{D_{3}}\left\|\partial _{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]1 \[\qquad\leq\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{4}(\mathbb{R})}^{4}+\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{ 3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{C(T)}{D_{3}}\left\|\partial _{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]2 \[\qquad\leq\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{4}(\mathbb{R})}^{4}+\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{ 3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{C(T)}{D_{3}}\left\|\partial _{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]3 \[\qquad\leq\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{4}(\mathbb{R})}^{4}+\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{ 3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{C(T)}{D_{3}}\left\|\partial _{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]4 \[\qquad\leq\frac{\gamma^{2}}{D_{3}}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{4}(\mathbb{R})}^{4}+\frac{\delta^{2}}{D_{3}}\left\|\partial_{x}^{ 3}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\frac{C(T)}{D_{3}}\left\|\partial _{x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]5Therefore, we have that  
\[ \begin{split}\frac{d}{dt}\left\|\partial_{x}^{3}u(t,\cdot)\right\|_{L ^{2}(\mathbb{R})}^{2}+2\beta^{2}\left\|\partial_{x}^{5}u(t,\cdot)\right\|_{L^{ 2}(\mathbb{R})}^{2}\\ =-6\gamma\int_{\mathbb{R}}\partial_{x}u\partial_{x}^{2}u\partial_ {x}^{5}udx+2\mu\left\|\partial_{x}^{4}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{ 2}-2\gamma\int_{\mathbb{R}}u\partial_{x}^{3}u\partial_{x}^{5}udx.\end{split}  \tag{53}\]  
Due to (21), (28), (49) and the Young inequality,  
\[ \begin{split} 6|\gamma|\int_{\mathbb{R}}|\partial_{x}u \partial_{x}^{2}u||\partial_{x}^{5}u|dx=& 6\int_{\mathbb{R}}\left|\frac{\gamma \partial_{x}u\partial_{x}^{2}u}{\beta\sqrt{D_{4}}}\right|\left|\beta\sqrt{D_{ 4}}\partial_{x}^{5}u\right|dx\\ \leq&\frac{3\gamma^{2}}{D_{4}}\int_{\mathbb{R}}( \partial_{x}u)^{2}(\partial_{x}^{2}u)^{2}dx+3\beta^{2}D_{3}\left\|\partial_{x }^{5}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\\ \leq&\frac{3\gamma^{2}}{\beta^{2}D_{4}}\left\| \partial_{x}u\right\|_{L^{\infty}((0,T)\times\mathbb{R})}^{2}\left\|\partial_{ x}^{2}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+3\beta^{2}D_{4}\left\| \partial_{x}^{5}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\\ \leq&\frac{C(T)}{D_{4}}+3\beta^{2}D_{4}\left\| \partial_{x}^{5}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2},\\ 2|\gamma|\int_{\mathbb{R}}|u\partial_{x}^{3}u||\partial_{x}^{5}u| dx=& 2\int_{\mathbb{R}}\left|\frac{\gamma u \partial_{x}^{3}u}{\beta\sqrt{D_{4}}}\right|\left|\beta\sqrt{D_{4}}\partial_ {x}^{5}udx\right|dx\\ \leq&\frac{\gamma^{2}}{\beta^{2}D_{4}}\int_{ \mathbb{R}}u^{2}(\partial_{x}^{3}u)^{2}dx+\beta^{2}D_{4}\left\|\partial_{x}^{ 5}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\\ \leq&\frac{\gamma^{2}}{\beta^{2}D_{4}}\left\|u \right\|_{L^{\infty}((0,T)\times\mathbb{R})}^{2}\left\|\partial_{x}^{3}u(t, \cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\beta^{2}D_{4}\left\|\partial_{x}^{5}u (t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\\ \leq&\frac{C(T)}{D_{4}}\left\|\partial_{x}^{3}u(t, \cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\beta^{2}D_{4}\left\|\partial_{x}^{5}u (t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2},\end{split} \]  
where \(D_{4}\) is a positive constant, which will be specified later. Therefore, by (53),  
\[