I am somewhat doubtful that the question as posed as any sort of reasonable answer. (Also, I don't really see how the Lorentzian metric even enter into the problem.)
(a) ANY hyperbolic PDE in (1+3) dimensions will, by definition, not have unique solution for your problem.
(b) Even elliptic PDEs are not guaranteed to have unique solutions, even for nice ones that come from minimization of a strictly convex energy functional: Euclidean space simply has too large a symmetry group, and if the equation itself is invariant under translations, any finite translate of a solution that decays at spatial infinity is another solution.
(c) If you restrict yourself to linear scalar PDEs $L[u] = 0$, by linearity, $u \equiv 0$ is necessarily a solution that satisfy your conditions. So your question as posed reduces to "which linear operator admits only the trivial solution". For constant coefficient $L$'s you can simply take the Fourier transform of both sides and study the zero-set of the PDO.
If you clarify your motivation for considering the question your are asking, maybe more reasonable answers can be given.