Let $\Omega$ be a bounded domain with smooth boundary and given $f \in H^{-1}((0,T)\times \Omega)$, consider the wave equation $$ \Box u =f\quad \text{on $(0,T)\times \Omega$},$$ with $u|_{(0,T)\times \partial \Omega}=0$ and $u(0,x)=\partial_t u(0,x)=0$ on $\Omega$.
Does this problem admit a unique solution in the energy space $$ u \in C(0,T;L^2(\Omega))\cap C^1(0,T;H^{-1}(\Omega)).$$
