Math 6410 • ODE

\textbf{Vibrating Plate.}
Let \(\Omega\subset\R^n\) be a smooth bounded domain. Let
\(\varphi,\psi\in C^\infty_0(\Omega)\). Suppose that there exists a solution
\(u(x,t)\in C^4(\overline{\Omega}\times[0,T])\) to the initial-boundary value problem for
the vibrating clamped plate equation
\[
\begin{cases}
  \dfrac{\partial^2u}{\partial t^2}+\Delta\Delta u=0,
    & \text{if }x\in\Omega\text{ and }t\ge0,\\
  u(x,0)=\varphi(x),\\
  \dfrac{\partial u}{\partial t}(x,0)=\psi(x),
    & \text{for }x\in\Omega,\\
  u(x,t)=\dfrac{\partial u}{\partial n}(x,t)=0,
    & \text{if }x\in\partial\Omega\text{ and }0\le t\le T.
\end{cases}
\tag{4.}
\]
\begin{enumerate}
\item[(a.)] Show that the total energy
\[
  \mathcal{E}(t)=\int_\Omega
  \left\{\left(\frac{\partial u}{\partial t}(x,t)\right)^2+
  \left(\Delta u(x,t)\right)^2\right\}\,dx
\]
is bounded in terms of \(\varphi\), \(\psi\) and \(t\).
\item[(b.)] Show that such a solution of (4.) is unique.
\end{enumerate}