The answer to the question if whether a solution $\chi$ to the the following equation exists
$$
-\frac{1}{c^{2}}\square=\left(\Delta - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\chi=f\;\text{ in }\;\Bbb R^4\equiv \Bbb R^3\times \Bbb R \label{w}\tag{W}
$$
under mild smoothness requirements on the datum $f$ is yes: I explain below why it is so in a constructive way, by actually constructing an explicit solution in two steps:
Construction of a fundamental solution: what is needed is a slightly modified fundamental solution of the D'Alembert operator, precisely the solution of the following equation:
$$
\square \mathscr{E}(x,t)=-c^2\delta(x,t)\label{da}\tag{DA}
$$
where $\delta(x,t)\equiv \delta(x)\times\delta(t)$ is the usual tensor product of Dirac measures respectively on the spatial and on the time domain. Once $\mathscr{E}(x,t)$ has been determined, we can find, provided certain compatiility conditions on $f$ are fulfilled (see below), a distributional solution $\chi(x,t)$ to the posed problem by convolution
$$
\chi(x,t)=\mathscr{E}\ast f(x,t)\label{s}\tag{S}
$$
The minimal requirements on $f$ is that the convolution product at the right term of \eqref{s} should exists as a distribution.
The regularity problem: prove that, provided $f$ is a "good" (for example $C^2$ smooth) function, the distribution $\chi$ in \eqref{s} is a "good" function in the same way.
Calculation of the modified fundamental solution for the D'Alembert operator in $\Bbb R^{3+1}$
We construct $\mathscr{E}$ as a distribution of slow growth (i.e. $\mathscr{E}\in \mathscr{S}^\prime$, see for example [1] §8.1-§8.2, pp. 113-116 or [2], §5.1-§5.2, pp. 74-78) by applying to PDE \eqref{da} the Fourier transform $\mathscr{F}_{x\to\xi}$ respect to the spatial variable $x$. By proceeding this way, \eqref{da} is transformed into the following ODE:
$$
\frac{\partial^2 \hat{\mathscr{E}}(\xi,t)}{\partial t^2} + c^2|\xi|^2\hat{\mathscr{E}}(\xi,t)=-c^2\delta(t)\label{1}\tag{1}
$$
Consider its equivalent standard form
$$
\frac{\partial^2 \hat{\mathscr{E}}_p(\xi,t)}{\partial t^2} + c^2|\xi|^2\hat{\mathscr{E}}_p(\xi,t)=\delta(t)\label{1'}\tag{1'}
$$
which has the same solutions, just multiplied by the constant $-c^2$: by solving it (see here, [1] §10.5, p. 147 or [2], §4.9, example 4.9.6 pp. 77-74 and §15.4, example 15.4.4) we get the following distribution
$$
\hat{\mathscr{E}}_p(\xi,t)= H(t)\frac{\sin c|\xi|t}{c|\xi|}\iff\hat{\mathscr{E}}(\xi,t)= -cH(t)\frac{\sin c|\xi|t}{|\xi|}\label{2}\tag{2}
$$
where $H(t)$ is the usual Heaviside function. Then, taking the inverse Fourier transform $\mathscr{F}_{\xi\to x}^{-1}\big(\hat{\mathscr{E}}\big)$ we get the sought for solution of \eqref{da} (see [1] §9.8, p. 135 and §10.7, p. 149)
$$
\mathscr{E}(x,t)=-\frac{H(t)}{4\pi t}\delta_{S_{ct}}(x)=-c\frac{H(t)}{2\pi }\delta\big(c^2t^2-|x|^2\big)\label{3}\tag{3}
$$
where
- $S_{ct}=\{x\in\Bbb R^3 | |x|^2=x_1^2+x_2^2+x_3^2=c^2t^2\}$ is the spherical light wave surface,
- $\delta_{S_{ct}}(x)$ is the Dirac measure supported on $S_{ct}$, otherwise called single layer measure.
Now, given any distribution $f\in\mathscr{D}(\Bbb R^{3+1})$ for which the convolution with $\mathscr{E}$ exists (for example any distribution of compact support) using \eqref{3} in formula \eqref{s} gives a generalized solution of \eqref{w}.
Construction of a regular solution
Instead of recurring to the standard (and complex) methods of regularity theory we will try a trickier way by looking carefully at the structure of \eqref{3} and on how this distribution acts on the space of infinitely smooth rapidly decreasing functions: precisely, given $\varphi\in\mathscr{S}$ we have that
$$
\begin{split}
\langle\mathscr{E},\varphi\rangle&=-\frac{1}{4\pi}\int\limits_{0}^{+\infty}\langle\delta_{S_{ct}},\varphi\rangle\frac{\mathrm{d}t}{t}\\
&=-\frac{1}{4\pi}\int\limits_{0}^{+\infty}\frac{1}{t}\int\limits_{S_{ct}}\varphi(x,t)\,\mathrm{d}\sigma_x\mathrm{d}t
\end{split}\label{4}\tag{4}
$$
From \eqref{4} we see that $\mathscr{E}$ acts on $\varphi\in\mathscr{S}$ as a spherical mean respect to the spatial $x\in \Bbb R^3$ variable and as a weighted time integral mean with weight function $t\mapsto {1\over t}\in L^1_\mathrm{loc}$ respect to the time variable $t\in\Bbb R_+$.
This implies that \eqref{4} is meaningful also for functions which are not in $\mathscr{S}$ nor are infinitely smooth. Precisely, provided that
- $\varphi(\cdot,t)\in L^1_\mathrm{loc}(\Bbb R^3)$ for almost all $t\in\Bbb R_+$, without any growth condition at infinity and
- $\varphi(x,\cdot)\in L^1_\mathrm{loc}(\Bbb R)$ with $|\varphi(x,t)|=O(t^{-\varepsilon})$ as $t\to\infty$ a.e. on $\Bbb R^3$ with $0<c\le\varepsilon$.
equation \eqref{4} is meaningful. Then, by putting
$$
\varphi(y,\tau)=f(x-y,t-\tau)
$$
and by using \eqref{4} jointly with the definition of convolution between a distribution and a function, i.e.
$$
\mathscr{E}\ast\varphi (x,t) \triangleq \langle \mathscr{E}, \varphi(x-y,t-\tau)\rangle
$$
we get the sought for solution
$$
\chi(x,t)=\mathscr{E}\ast f(x,t)=-\frac{1}{4\pi}\int\limits_{0}^{+\infty}\frac{1}{\tau}\int\limits_{S_{c\tau}}f(x-y,t-\tau)\,\mathrm{d}\sigma_y\mathrm{d}\tau
\label{S}\tag{WS}
$$
Notes
- The hypothesis $n=3$, i.e. the fact that we are working in a $3$D space, is essential for defining the structure of \eqref{3}. The inverse transform of $\hat{\mathscr{E}}$ in \eqref{2} has not the same structure on every $\Bbb R^n$: in monographs on hyperbolic PDEs, this concept is also stated by saying that Huygens's principle does not hold in even spatial dimension.
- The regularity of the solution we have obtained is very weak: in particularly we do not know the smoothness of $\chi$ for a given smoothness of $f$. Deeper methods are required for the investigation of this problems.
[1] V. S. Vladimirov (1971)[1967], Equations of mathematical physics, Translated from the Russian original (1967) by Audrey Littlewood. Edited by Alan Jeffrey, (English), Pure and Applied Mathematics, Vol. 3, New York: Marcel Dekker, Inc., pp. vi+418, MR0268497, Zbl 0207.09101.
[2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029.
Best Answer
I finally managed to find a nice reference specifically for the three-dimensional situation: