Does the wave equation $u\_{tt} - \Delta u = 0$ have any backward uniqueness results that are similar to the ones for the heat equation (see for example Theorem 11 page 64 in Evans)? If not, are there any counterexamples?



Backward Uniqueness for the wave equation
Suppose $u$ solves
\begin{cases}
(\partial\_t^2-\Delta)u = 0 & \text{on } U\\
u(T,x) = 0 \\
\partial\_t 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.


