Consider the PDE with boundary conditions:

\begin{align*}

&u_{tt}+K u_{xxxx} + \omega^2 u =0,\qquad&&0<x<L,\\[3pt]

&u(0,t)=u_x(0,t)=0,\\[3pt]

&u(L,t)=u_x(L,t)=0,

\end{align*}

where $K>0$ is constant. Prove that the energy $E(t)$ defined as

\begin{equation*}

E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + K u_{xx}^2 + \omega^2 u^2)\,dx

\end{equation*}

does not depend on $t$.