# On Classical Solutions for A Kuramoto-Sinelshchikov-Velarde-Type Equation
## 2 Proof of Theorem 1, under the Assumptions (2) and (4)  
In this section, we prove Theorem 1, under the assumptions (2) and (4). Thanks to (2), (1) reads  
\[ \begin{cases}\partial_{t}u+\kappa(\partial_{x}u)^{2}+q(\partial_{x}u)^{3}+r( \partial_{x}u)^{4}+\delta\partial_{x}^{3}u+\beta^{2}\partial_{x}^{4}u+\mu \partial_{x}^{2}u_{c},&t>0,\quad x\in\mathbb{R},\\ u(0,x)=u_{0}(x),&x\in\mathbb{R}.\end{cases}  \tag{17}\]  
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 \(T\). We prove the following lemma. Lemma 1: _Fix \(T>0\). Then, we have that_  
\[\|\partial_{x}u(t,\cdot)\|_{L^{2}(\mathbb{R})}^{2}+e^{\frac{p^{2}t}{\beta^{2} }}\beta^{2}\int_{0}^{t}e^{-\frac{p^{2}t}{\beta^{2}}}\left\|\partial_{x}^{3}u( s,\cdot)\right\|_{L^{2}(\mathbb{R})}^{2}ds\leq C(T), \tag{18}\]  
The proof of the previous lemma is based on the regularity of the functions \(u\) and the following result. **Lemma 3**: _For each \(t\geq 0\), we have that_  
\[\int_{\mathbb{R}}|u||\partial_{x}u|^{3}dx\leq 2\sqrt{\|u(t,\cdot)\|_{L^{2}(\mathbb{R})}}\sqrt{\|\partial_{x}^{2} u(t,\cdot)\|_{L^{2}(\mathbb{R})}}\left\|\partial_{x}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{3}, \tag{23}\] \[\int_{\mathbb{R}}|u||\partial_{x}u|^{4}dx\leq 2\sqrt{\|u(t,\cdot)\|_{L^{2}(\mathbb{R})}}\left\|\partial_{x}^{2} u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}\sqrt{\|\partial_{x}u(t,\cdot)\|_{L^{2}( \mathbb{R})}^{2}}. \tag{24}\]  
Proof. : We begin by proving (24). Thanks to the regularity of the function \(u\) and the Holder inequality,  
\[u^{2}(t,x)=2\int_{-\infty}^{x}u\partial_{x}udy\leq 2\int_{\mathbb{R}}|u|| \partial_{x}u|dx\leq 2\left\|u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}\left\| \partial_{x}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}.\]  
Therefore,  
\[\left\|u(t,\cdot)\right\|_{L^{\infty}(\mathbb{R})}^{2}\leq 2\left\|u(t,\cdot) \right\|_{L^{2}(\mathbb{R})}\left\|\partial_{x}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}. \tag{25}\]  
Again, by the regularity of the function \(u\) and the Holder inequality,  
\[(\partial_{x}u(t,x))^{2}=2\int_{-\infty}^{x}\partial_{x}u\partial_{x}^{2}udy \leq 2\int_{\mathbb{R}}|\partial_{x}u||\partial_{x}^{2}u|dx\leq 2\left\| \partial_{x}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}\left\|\partial_{x}^{2}u(t, \cdot)\right\|_{L^{2}(\mathbb{R})}.\]  
Hence,  
\[\left\|\partial_{x}u(t,\cdot)\right\|_{L^{\infty}(\mathbb{R})}^{2}\leq 2\left\| \partial_{x}u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}\left\|\partial_{x}^{2}u(t, \cdot)\right\|_{L^{2}(\mathbb{R})}. \tag{26}\]  
Consequently, by (25) and (26),  
\[\int_{\mathbb{R}}|u||\partial_{x}u|^{3}dx\leq \left\|u(t,\cdot)\right\|_{L^{\infty}(\mathbb{R})}\left\|\partial _{x}u(t,\cdot)\right\|_{L^{\infty}(\mathbb{R})}\left\|\partial_{x}u(t,\cdot) \right\|_{L^{2}(\mathbb{R})}^{2}\] \[\leq 2\sqrt{\left\|u(t,\cdot)\right\|_{L^{2}(\mathbb{R})}}\sqrt{\| \partial_{x}^{2}u(t,\cdot)\|_{L^{2}(\mathbb{R})}}\left\|\partial_{x}u(t, \cdot)\right\|_{L^{2}(\mathbb{R})}^{3},\]  
that is (23). Finally, we prove (24). By (25) and (26),  
\[\int_{\mathbb{R}}|u||\partial_{x}u|^{4}dx\leq \left\|u(t,\cdot)\right\|_{L^{\infty}(\mathbb{R})}\left\|\partial _{x}u(t,\cdot)\right\|_{L^{\infty}(\mathbb{R})}^{2}\left\|\partial_{x}u(t, \cdot)\right\|_{L^{2}(\mathbb{R})}^{2}\] \[\int_{\mathbb{R}}|u||\partial_{x}u|^{3}dx\leq 2\sqrt{\|u(t,\cdot)\|_{L^{2}(\mathbb{R})}}\sqrt{\|\partial_{x}^{2} u(t,\cdot)\|_{L^{2}(\mathbb{R})}}\left\|\partial_{x}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{3}, \tag{23}\]0  
which gives (24). Proof of Lemma 2.: Let \(0\leq t\leq T\).