Suppose $u$ solves
\begin{cases}
(\partial_t^2-\Delta)u = 0 & \text{on } U\\
u(T,x) = 0 \\
u(t,x) = 0 & \text{on } \partial U.
\end{cases}
Since the energy 
$$
E(t) = \int_U |\nabla u|^2 + |\partial_t u|^2\,dx.
$$
is constant, $E(t) = E(T) = 0$, it follows that $u \equiv 0$. From linearity of the wave equation, the backwards uniqueness result follows. In general, solving the wave equation backwards in time is not really different from solving it forwards in time.