# On Classical Solutions for A Kuramoto-Sinelshchikov-Velarde-Type Equation
## 2 Proof of Theorem 1, under the Assumptions (2) and (4)
\begin{split} 2|\kappa|&\int_{\mathbb{R}}|( \partial_{x}u_{1}+\partial_{x}u_{2})||\omega||\partial_{x}\omega|dx\\ &\qquad\leq\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2 }+\kappa^{2}\int_{\mathbb{R}}(\partial_{x}u_{1}+\partial_{x}u_{2})^{2}( \partial_{x}\omega)^{2}dx\\ &\qquad\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|q|&\int_{\mathbb{R}}|(\partial_{x}u_{1})^{2}+( \partial_{x}u_{2})^{2}+\partial_{x}u_{1}\partial_{x}u_{2}||\omega||\partial_{ x}\omega|dx\\ &\qquad\leq\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2 }+q^{2}\int_{\mathbb{R}}[(\partial_{x}u_{1})^{2}+(\partial_{x}u_{2})^{2}+ \partial_{x}u_{1}\partial_{x}u_{2}]^{2}(\partial_{x}\omega)^{2}dx\\ &\qquad\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|r|&\int_{\mathbb{R}}|(\partial_{x}u_{1})^{2}+( \partial_{x}u_{2})^{2}||\partial_{x}u_{1}+\partial_{x}u_{2}||\omega||\partial_{ x}\omega|dx\\ &\qquad\leq\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2 }+r^{2}\int_{\mathbb{R}}((\partial_{x}u_{1})^{2}+(\partial_{x}u_{2})^{2})^{2}( \partial_{x}u_{1}+\partial_{x}u_{2})^{2}(\partial_{x}\omega)^{2}dx\\ &\qquad\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}.\end{split} \]  
It follows from (40) that  
\[ \begin{split}\frac{d}{dt}\left\|\omega(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}+2\beta^{2}\left\|\partial_{x}^{2}\omega(t,\cdot)\right\|_{L^ {2}(\mathbb{R})}^{2}\\ \leq& 3\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2 }+C(T)\left\|\partial_{x}\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\end{split}  \tag{42}\]  
Due to the Young inequality,  
\[\begin{split} C(T)\left\|\partial_{x}\omega(t,\cdot)\right\|^{2}=-C( T)\int_{\mathbb{R}}\omega\partial_{x}^{2}\omega dx\leq& C(T)\int_{\mathbb{R}}|\omega||\partial_{x}^{2}\omega|dx=2\int_{ \mathbb{R}}\left|\frac{C(T)\omega}{2\beta}\right|\left|\beta\partial_{x}^{2} \\[\leq C(T)\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\beta^{2} \left\|\partial_{x}^{2}\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]  
Therefore, by (42),  
\[\frac{d}{dt}\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\beta^{2} \left\|\partial_{x}^{2}\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\leq C(T )\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}.\]  
The Gronwall Lemma and (37) gives  
\[\left\|\omega(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\beta^{2}e^{C(T)t} \int_{0}^{t}e^{-C(T)s}\left\|\partial_{x}^{2}\omega(s,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2}ds\leq e^{C(T)t}\left\|\omega_{0}\right\|_{L^{2}(\mathbb{R})}. \tag{43}\]  
Equation (16) follows from (36) and (43). **3. Proof of Theorem 1, under the assumptions (3) and (4) or (5)**  
In this section, we prove Theorem 1, under the assumptions (3) and (4). Thanks to (3), (1) reads  
\[ \begin{cases}\partial_{t}u+2\gamma(\partial_{x}u)^{2}+\delta\partial_{x}^{3}u +\beta^{2}\partial_{x}^{4}u+\mu\partial_{x}^{2}u_{\epsilon}+\gamma u\partial_ {x}^{2}u=0,&t>0,\quad x\in\mathbb{R},\\ u(0,x)=u_{0}(x),&x\in\mathbb{R},\end{cases}  \tag{44}\]  
Let us prove some a priori estimates on \(u\). We denote with \(C_{0}\) the constants which depend only on the initial data, and with \(C(T)\), the constants which depend also on \(T\). **Lemma 6**. : _Fix \(T>0\) and assume (4) or (5). There exists a constant \(C(T)>0\), such that_  
\[\left\|u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}+\beta^{2}e^{\frac{t^{2}t^{2} }{\beta^{2}}}\int_{0}^{t}e^{-\frac{t^{2}\beta^{2}}{\beta^{2}}}\left\|\partial _{x}^{2}u(s,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}ds\leq C(T), \tag{45}\]  
_for every \(0\leq t\leq T\)._  
Proof. : Let \(0\leq t\leq T\).