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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).

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    $\begingroup$ Not even the ODE $y'=0$ has a unique solution. Additional conditions are needed, for the wave equation they are called initial conditions. If you add them, the behaviour at infinity has no influence. Maybe you should read some introductory course on this topic $\endgroup$
    – Piero D'Ancona
    Commented May 19, 2020 at 13:41
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    $\begingroup$ This is treated in most PDE textbooks; Evans' textbook has it in one of the early chapters. The solution operator for the inhomogeneous problem can be constructed from that of the homogeneous, initial value problem using Duhamel's principle. // Alternatively, I think it is also treated in Chapter 1 of Sogge's Nonlinear Wave Equations and one of the early chapters of Shatah and Struwe's Geometric Wave Equations. $\endgroup$
    – Willie Wong
    Commented May 19, 2020 at 14:20
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    $\begingroup$ RE your edit: the "initial condition" used here is actually a scattering type condition. It is saying that the solution has no incoming radiation from past (null) infinity. For $\rho$ with sufficiently fast space-time decay, this solution can be picked out as the unique one such that $\lim_{v \to \infty} vu(t - v, x - \omega v) = 0$ (for all $t,x,\omega$; here $\omega \in \mathbb{S}^2$). I think the keyword to search for is "Sommerfeld radiation condition". $\endgroup$
    – Willie Wong
    Commented May 19, 2020 at 15:57
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    $\begingroup$ @PieroD'Ancona: I try to read the tea-leaves and guess what MKO is looking for based on what little I remember of Feynman's lectures. When $\rho$ has compact space-time support, every solution to the wave equation can be written as the sum of the formula that Feynman gave plus a free wave. If I squint and pretend MKO remembered incorrectly about "decay and spatial infinity" and in fact is referring to "some asymptotic condition", then the far field condition (you can read it as "size $o(v^{-1})$ relative to null foliation") does the job. (Yes, lots of guess work. So comment, not answer.) $\endgroup$
    – Willie Wong
    Commented May 19, 2020 at 17:53
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    $\begingroup$ i think that the following link will be useful for the purposes of the OP: farside.ph.utexas.edu/teaching/jk1/lectures/node7.html $\endgroup$
    – Konstantinos Kanakoglou
    Commented May 20, 2020 at 2:37

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