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?
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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.
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5$\begingroup$ Exactly the same, in fact: if $u(x,t)$ is a solution, so is $u(x,-t)$. $\endgroup$ Commented Apr 28, 2015 at 2:19
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1$\begingroup$ But what if the boundary conditions are different? There are actually nontrivial problems (even unsolved problems) here. I doubt, however, that this is what the OP had in mind. $\endgroup$ Commented Apr 28, 2015 at 14:07
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