# On Classical Solutions for A Kuramoto-Sinelshchikov-Velarde-Type Equation
## 2 Proof of Theorem 1, under the Assumptions (2) and (4)
It follows from (35) that  
\[\leq\frac{\kappa^{2}}{D_{2}}\left\|\partial_{x}u(t,\cdot)\right\|_ {L^{4}(\mathbb{R})}^{4}+D_{2}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2},\]4 \[\leq\frac{\kappa^{2}}{D_{2}}\left\|\partial_{x}u(t,\cdot)\right\|_ {L^{4}(\mathbb{R})}^{4}+D_{2}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2},\]5  
Choosing \(D_{2}=\frac{1}{2}\), we have that  
\[\leq\frac{\kappa^{2}}{D_{2}}\left\|\partial_{x}u(t,\cdot)\right\|_ {L^{4}(\mathbb{R})}^{4}+D_{2}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2},\]6 \[\leq\frac{\kappa^{2}}{D_{2}}\left\|\partial_{x}u(t,\cdot)\right\|_ {L^{4}(\mathbb{R})}^{4}+D_{2}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2},\]7  
Integrating on \((0,t)\), by (4), (18), (20) and (22), we obtain  
\[\leq\frac{\kappa^{2}}{D_{2}}\left\|\partial_{x}u(t,\cdot)\right\|_ {L^{4}(\mathbb{R})}^{4}+D_{2}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2},\]8 \[\leq\frac{\kappa^{2}}{D_{2}}\left\|\partial_{x}u(t,\cdot)\right\|_ {L^{4}(\mathbb{R})}^{4}+D_{2}\left\|\partial_{t}u(t,\cdot)\right\|_{L^{2}( \mathbb{R})}^{2},\]9\[2|q|\int_{\mathbb{R}}|\partial_{x}u|^{3}|\partial_{t}u|dx= 2\int_{\mathbb{R}}\left|\frac{q(\partial_{x}u)^{3}}{\sqrt{D_{2 }}}\right|\Big{|}\sqrt{D_{2}}\partial_{t}u\Big{|}\,dx\]0 \[2|q|\int_{\mathbb{R}}|\partial_{x}u|^{3}|\partial_{t}u|dx= 2\int_{\mathbb{R}}\left|\frac{q(\partial_{x}u)^{3}}{\sqrt{D_{2 }}}\right|\Big{|}\sqrt{D_{2}}\partial_{t}u\Big{|}\,dx\]1 \[2|q|\int_{\mathbb{R}}|\partial_{x}u|^{3}|\partial_{t}u|dx= 2\int_{\mathbb{R}}\left|\frac{q(\partial_{x}u)^{3}}{\sqrt{D_{2 }}}\right|\Big{|}\sqrt{D_{2}}\partial_{t}u\Big{|}\,dx\]2  
which gives (34). Now, we prove Theorem 1. Proof of Theorem 1.: Fix \(T>0\). Thanks to Lemmas 1, 2, 4 and the Cauchy-Kovalevskaya Theorem [50], we have that \(u\) is solution of (1) and (14) holds. We prove (16). Let \(u_{1}\) and \(u_{2}\) be two solutions of (1), which verify (14) that is  
\[ \begin{cases}\partial_{t}u_{1}+\kappa(\partial_{x}u_{1})^{2}+q( \partial_{x}u_{1})^{3}+r(\partial_{x}u_{1})^{4}+\delta\partial_{x}^{3}u_{1}+ \mu\partial_{x}^{2}u_{1}+\beta^{2}\partial_{x}^{4}u_{1}=0,&t>0,\,x\in\mathbb{ R},\\ u_{1}(0,x)=u_{1,0}(x),&x\in\mathbb{R},\\ \end{cases} \] \[ \begin{cases}\partial_{t}u_{2}+\kappa(\partial_{x}u_{2})^{2}+q( \partial_{x}u_{2})^{3}+r(\partial_{x}u_{2})^{4}+\delta\partial_{x}^{3}u_{2}+ \mu\partial_{x}^{2}u_{2}+\beta^{2}\partial_{x}^{4}u_{2}=0,&t>0,\,x\in\mathbb{ R},\\ u_{1}(0,x)=u_{1,0}(x),&x\in\mathbb{R},\end{cases} \]  
Then, the function  
\[\omega=u_{1}-u_{2} \tag{36}\]  
is the solution of the following Cauchy problem:  
\[ \begin{cases}\partial_{t}\omega+\kappa\left[(\partial_{x}u_{1})^{2}-(\partial _{x}u_{2})^{2}\right]+q\left[(\partial_{x}u_{1})^{3}-(\partial_{x}u_{2})^{3} \right]\\ \qquad\qquad+r\left[(\partial_{x}u_{1})^{4}-(\partial_{x}u_{2})^{4}\right]+ \delta\partial_{x}^{3}\omega\\ \qquad\qquad+\mu\partial_{x}^{2}\omega+\beta^{2}\partial_{x}^{4}\omega=0,&t>0, \,x\in\mathbb{R},\\ \omega(0,x)=u_{1,0}(x)-u_{2,0}(x),&x\in\mathbb{R}.\end{cases}  \tag{37}\]  
Observe that thanks to (36),  
\[(\partial_{x}u_{1})^{2}-(\partial_{x}u_{2})^{2}= (\partial_{x}u_{1}+\partial_{x}u_{2})(\partial_{x}u_{1}-\partial _{x}u_{2})=(\partial_{x}u_{1}+\partial_{x}u_{2})\partial_{x}\omega,\] \[(\partial_{x}u_{1})^{3}-(\partial_{x}u_{2})^{3}= [(\partial_{x}u_{1})^{2}+(\partial_{x}u_{2})^{2}+\partial_{x}u_{1} \partial_{x}u_{2}](\partial_{x}u_{1}-\partial_{x}u_{2})\] \[= [(\partial_{x}u_{1})^{2}+(\partial_{x}u_{2})^{2}+\partial_{x}u_{1 }\partial_{x}u_{2}]\partial_{x}\omega,\] \[(\partial_{x}u_{1})^{4}-(\partial_{x}u_{2})^{4}= [(\partial_{x}u_{1})^{2}+(\partial_{x}u_{2})^{2}][(\partial_{x}u_{1 })^{2}-(\partial_{x}u_{2})^{2}]\] \[= [(\partial_{x}u_{1})^{2}+(\partial_{x}u_{2})^{2}](\partial_{x}u_{1 }+\partial_{x}u_{2})(\partial_{x}u_{1}-\partial_{x}u_{2})\] \[= [(\partial_{x}u_{1})^{2}+(\partial_{x}u_{2})^{2}](\partial_{x}u_{1 }+\partial_{x}u_{2})\partial_{x}\omega.\]  
Consequently, (37) is equivalent to the following equation:  
\[