I'm trying to prove the uniqueness of solutions of Euler - Bernoulli beam equation:
$\left\{\begin{matrix} u_{tt}+a^{2}u_{xxxx}=0, &0<x<L \\ u(x,0)=0, &0\le x\le L\\ u_{t}(x,0)=0, &0\le x\le L \end{matrix}\right.$
My attempt: I multiply the equation by $u_{t}$, then integrating and defining the energy of the equation $$E(t)=\dfrac{1}{2}\int_{0}^{L}[u_{t}^{2}+a^{2}u_{xx}^{2}]dx.$$ However, $\dfrac{dE}{dt}=a^{2}\left[u_{xt}u_{xx}-u_{t}u_{xxx}\right]\Big|_{0}^{L}$ so we only have $E'(0)=0$ and now I don't know how to prove that $E(t)$ is constant. It seems like the energy functional should be different but I haven't figured out yet.
Using the equality of mixed partial derivatives, integration by parts and the PDE itself, we have indeed $\frac{\text d}{\text d t}E =a^2[u_{tx}u_{xx}-u_{t}u_{xxx}]_0^L$. This derivative is equal to zero if ad hoc boundary conditions are applied. Then, the energy method provides uniqueness. But here, boundary conditions are missing.