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 (say $\rho$ is compactly supported would be fine). 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).

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).

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.

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 (say $\rho$ is compactly supported would be fine). 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).