In the domain $(0,1) \times [0,T], T>0$, we consider the boundary value problem
$$V_{tt} + \eta V = (\xi V_x - \beta V_{xxx})_x,\,\,\,\,\,V(0,t)=0, V(1,t)=0, V_x(0,t)=0, V_x(1,t)=0,$$
where $\eta, \xi, \beta$ are real positive constants. Find the energy of the solution to this problem and show that it is conserved.
My Solution I multiplied by $V_t$ and integrated it with respect to the variable $x$. I ended up with the expression:
$$\frac{d}{dt} \left( \frac{1}{2} \int_0^1 [v^2 + v_t^2 +\xi v_x^2 +\beta v_{xx}^2]dx \right)=0.$$
Then I defined the energy functional as:
$$E(t) = \frac{1}{2} \int_0^1 [v^2 + v_t^2 +\xi v_x^2 +\beta v_{xx}^2]dx.$$
Is my solution correct?