The question is posed specifically about the wave equation and is related to partial hypoellipticity of the wave equation. Let $\Omega \subset \mathbb R^n$, $M=(0,T)\times \Omega$ and $\Gamma=(0,T)\times \partial \Omega$. Suppose that $u \in L^2(M)$ satisfies $\Box u=0$ on $M$ in the distributional sense. Is the trace of $u$ well defined on $\Gamma$ as an element in $L^2(\Gamma)$? Similarly, is the trace $u|_{t=0}$ and $\partial_t u|_{t=0}$ well defined as elements in $L^2(\Omega)$ and $H^{-1}(\Omega)$ respectively?
Note that Sobolev trace theorems do not imply what I am asking for, but because $u$ solves the wave equation, I believe this trace is well-defined.
