Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \Box u+qu=0\,\quad &\text{on $(0,\infty)\times \Omega$}, \\ u=f\,\quad &\text{on $\Sigma=(0,\infty)\times \partial \Omega$,}\\ (u,\partial_t u)\to 0 \,\quad &\text{ on $\Omega$ as $t\to \infty$} \end{cases} \end{aligned} \end{equation}

Suppose that $q \in C^{\infty}([0,\infty)\times \overline{\Omega})$ and that $f \in C^{\infty}_c((0,\infty)\times \partial \Omega)$. Does the above boundary value problem admit a unique smooth solution?

Note that if initial data was assumed to be zero, this problem would be trivial by finite speed of propagation. However, I wonder if a similar technique could work when data vanishes at infinity instead. Uniqueness is the key issue here as existence is easy.