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.

**ADDED:** Let me state my question more precisely. In Feynman's lectures in physics, Ch. 21 $\S$ 3, there is given a solution of the above wave equation for $n=3$ as follows (in different notation):
$$u(x,t)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{\rho(y,t-|x-y|)}{|x-y|}dy.$$
It is implicity assumed  that the integral converges. To discard other solutions Feynman appeals to physical intuition. **I am wondering which mathematical conditions should be imposed on the solutions in order to get the above solution only.** Here no initial conditions are used apparently, only some decay at infinity (but it is not clear to me exactly).