Consider the inhomogeneous wave equation $$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$ where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\rho$ is a given function.

This equation has non-unique solutions.

However I have heard that if one assumes in addition some decay of a solution at spatial infinity then solution becomes unique and is given by an explicit formula. I would be interested to have a precise statement with all the assumptions and a reference. The case $n=3$ is particularly interesting to me.

I guess this material should be very standard, but I am not a specialist.