I have been reading up on hyperboloidal compactifications here. I am thinking about scalar wave equations of the form (context I am working in four dimensional spacetime (three space dimensions, one time), with metric signature -+++, and assuming an asymptotically flat spacetime):
\begin{equation} \Box\phi - m^2\phi - \lambda \phi^3 = 0, \end{equation} where $m^2$ and $\lambda$ are constants. Looking at the techniques in this paper, the author says the it's not yet known if one can hyperboloidally compactify the above wave equation when $m^2\neq0$, while having the equation be regular at future null infinty. Looking at the citations, I haven't found any progress on this.
My question: Is there any work on regular hyperboloidal compactifications (i.e. the equation itself is regular at future null infinity) of the massive scalar wave equation?
More background:
Consider Minkowski space, then the wave equation is \begin{align} \left(-\partial_t^2 + \partial_r^2 + \frac{2}{r}\partial_r + \frac{1}{r^2}\Delta_{S^2}\right)\phi - m^2\phi - \lambda\phi^3=0 . \end{align} By rescaling $\phi=r^{-1}\psi$ and multiplying by $r$, we get \begin{align} \left(-\partial_t^2 + \partial_r^2 + \frac{1}{r^2}\Delta_{S^2}\right)\psi - m^2\psi - \frac{1}{r^2}\lambda\psi^3=0 . \end{align} Following 1, we compactify by setting \begin{equation} t = T - f(r), \qquad r = \frac{R}{1-R} \end{equation} where $f$ needs to satisfy $\lim_{R\to1} \frac{df}{dr} \sim 1 - (1-R)^2$. It turns out that when we do this the mass term goes as $m^2\psi/(1-R)^2$ in the compactified equations of motion. My hope is that there is a way of hyperboloidally compactifying the equation that avoids this problem.
The singularity of the mass term at null infinity is independent of the foliation. See Winicour (1988) and Helfer (1993). One may nevertheless speculate that under certain special scenarios the compactification of the Klein-Gordon equation at null infinity may lead to a regular equation. And this is indeed the case.
Write the wave equation in frequency domain via $\psi = e^{ikt} \Psi$. Ignoring the cubic term $$ \left(\partial_r^2 + \frac{1}{r^2}\Delta_{S^2} \right) \Psi + (k^2-m^2) \Psi = 0. $$
There is no time in frequency domain, so to transform this equation into hyperboloidal coordinates you simply rescale the unknown $\Psi$ with $e^{-ikf(r)}$ as described here. The trick in the massive case is to change the rescaling to $e^{-i\sqrt{k^2-m^2} f(r)}$. After compactification, you'll see that the equation is regular at null infinity for each frequency.
This is as yet unpublished, but maybe it's worth publishing.