I will start this one.
We take the time derivative of the energy as given in the problem. So: \begin{equation}
\frac{1}{2}\int_0^{\infty}[(2u_tu_{tt} + 2ku_{xx}u_{xxt}]dx = \int_0^{\infty}[u_t(-ku_{xxxx}) + ku_{xx}u_{xxt}]dx \end{equation}
Now we will integrate by parts twice on the second term in this integral. I will omit the steps and state the result: \begin{equation}
-k\int_0^{\infty}u_tu_{xxxx}dx +ku_{xx}u_{xt}|_0^{\infty} - ku_tu_{xxx}|_0^{\infty} + k\int_0^{\infty}u_tu_{xxxx}dx \end{equation}
The first and last terms cancel. Let's make use of the first boundary condition. \begin{equation}
u|_{x=0} = u_x|{x=0} = 0 \longrightarrow u_t|_{x=0} = u_{xt}|_{x=0} = 0 \end{equation}
We plug this in to get the final result: \begin{equation}
\frac{\partial{}E(t)}{\partial{}t} = 0 \end{equation}
I will add the other BCs later; or if someone wants to collaborate and add those ones please do!
