TITLE: Green's function for fourth order equation
QUESTION [5 upvotes]: I know the D'Alembert operator ${\frac {1}{c^{2}}}\partial _{t}^{2}-\Delta _{\text{3D}}$ has a well-known Green's function $\frac{\delta(t-\frac{r}{c})}{4 \pi r}$. This is very useful for studying 3D wave equation / fluids.
How about the Green's function of the  following operator?
$${\frac {1}{\mu^{4}}}\partial _{t}^{2}+\Delta _{\text{3D}}\Delta _{\text{3D}}$$
Is it known? How do I find it? This would be useful for understanding superfluids.

REPLY [3 votes]: The time-independent equation is the biharmonic equation,
$$-\Delta^2f({\bf r},{\bf r}')=\delta({\bf r}-{\bf r}'),$$
with 3D solution
$$f({\bf r},{\bf r}')=\frac{1}{8\pi}|{\bf r}-{\bf r}'|,$$
see Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation (2012). More generally, the Green function of the $k$-the power of the Laplacian, $(-\Delta)^k$ is $\propto |{\bf r}-{\bf r}'|^{2k-d}$ in odd dimensions $d$.
The time-dependent case has been studied in 1D in On the biharmonic wave equation (1967) and in higher dimensions in A uniqueness condition for the polyharmonic equation in free space (1990).[The first paper gives the 1D Green function; I don't have access to the second paper, so I don't know if it contains the 3D Green function.]