Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $(+-\cdots-)$, i.e. the signature of $g$ is $2-d$) with Levi-Civita connection $\nabla$, and consider the Klein-Gordon equation on $(M,g)$: $$\tag{1}\label{e1}\Box_g u+m^2 u=0\ ,\quad\Box_g=g^{-1}\nabla^2\ .$$ We assume $(M,g)$ to be globally hyperbolic: there is $\tau\in\mathscr{C}^\infty(M,\mathbb{R})$ surjective such that $\mathrm{d}\tau$ is a future-directed, timelike covector field and $\Sigma_t=\tau^{-1}(t)$ is a Cauchy hypersurface for $(M,g)$ (i.e. $M=$ the Cauchy development of $\Sigma_t$ in $(M,g)$) for all $t\in\mathbb{R}$. We then say that $\tau$ is a Cauchy time function for $(M,g)$, and one can show that $M$ is then diffeomorphic to $\mathbb{R}\times\Sigma_t$ for every $t\in\mathbb{R}$ (it suffices to follow the maximal orbits of a future-directed timelike vector field $T$ - e.g. $T=g^\sharp(\mathrm{d}\tau)$ -, which must intersect $\Sigma_t$ for all $t\in\mathbb{R}$ since they are all Cauchy hypersurfaces by hypothesis). We can (and shall) use a Cauchy time function $\tau$ to provide us a global time coordinate.
If $(M,g)=(\mathbb{R}^d,\eta)$ is Minkowski space-time with $\tau=$ the standard (Cartesian) time coordinate $x^0$, the stationary phase method (among others) allows us to show that if $u$ is a smooth solution of (1) with initial data of compact support and $m\neq 0$, then $$\tag{2}\label{e2}\|u(x^0,\cdot)\|_{L^\infty(\mathbb{R}^{d-1})}=O\Big(|x^0|^{-\frac{d-1}{2}}\Big)\ .$$ If $m=0$, we have instead $$\tag{3}\label{e3}\|u(x^0,\cdot)\|_{L^\infty(\mathbb{R}^{d-1})}=O\Big(|x^0|^{-\frac{d-2}{2}}\Big)\ .$$
On the other hand, if $M=\mathbb{R}\times\mathbb{S}^{d-1}\ni(t,\theta)$ and $g=\mathrm{d}t\otimes\mathrm{d}t-h$, where $h$ is the standard round metric on $\mathbb{S}^{d-1}$ (i.e. $(M,g)=$ the so-called Einstein cylinder or Einstein static universe), then in both cases we have no decay at all. Intuitively, this is due to the fact that all light rays are trapped in a compact spatial region due to the simple fact that $(M,g)$ has compact Cauchy hypersurfaces. For cases in between (e.g. Schwarzschild and Kerr geometries, etc.) with non-compact Cauchy hypersurfaces (which is the only case I am interested in from now on), there are two possible (not completely independent) phenomena due to focusing of null geodesics which slow down global decay of solutions of \eqref{e1} as compared with \eqref{e2} and \eqref{e3}:
Trapping. This means that at least some null geodesics remain within a compact spatial region (often lower-dimensional). This is the case of static black hole space-times such as Schwarzschild - light rays with initial position at the radius $r=3M$ (in $d=4$) and purely angular initial velocity stay forever in the photosphere $r=3M$;
Caustics. This happens whenever (say, future-directed) null geodesics starting at a point $p\in M$ go beyond the (future) null conjugate locus of $p$.
Typically, trapping has a stronger effect in reducing global decay than caustics. On the other hand, caustics are usually stable against small perturbations of $g$, whereas trapping need not be. A lot of work has been dedicated by several people (Alinhac, Baskin, Sogge, Tataru, etc.) to investigate decay of solutions of \eqref{e1} when $(M,g)$ is either free of 1. and 2. or has a particular kind of trapping.
Having set up our context and framework, I am now finally able to pose my question. Consider the Klein-Gordon equation \eqref{e1} in a globally hyperbolic $(M,g)$ endowed with a steep Cauchy time function $\tau$ (i.e. $g^{-1}(\mathrm{d}\tau,\mathrm{d}\tau)\geq 1$). Since the set of all globally hyperbolic smooth Lorentzian metrics on $M$ is Whitney-$\mathscr{C}^0$-open in the space of all smooth Lorentzian metrics on $M$, we are able to perform small perturbations $g'$ of $g$ in that topology while keeping the globally hyperbolic character of $g'$. Moreover, if the perturbation is suitably small in that topology, a steep Cauchy time function remains a Cauchy time function with respect to $g'$.
Question: Suppose that smooth solutions to \eqref{e1} in $(M,g)$ with compactly supported initial data in $\Sigma_0$ have a time decay rate of the form $$\|u\|_{L^\infty(\Sigma_t)}=O(f(|t|))\ ,\quad t>0$$ for some $\delta>0$, where $f:(0,+\infty)\to(0,+\infty)$ is a strictly decreasing function satisfying $\lim_{t\to+\infty}f(t)=0$. Assume that the Cauchy hypersurfaces of $(M,g)$ are not compact. For a "generic" small perturbation $g'$ within the above conditions, what is the time decay rate for solutions $u'$ of the same kind to $$\tag{1'}\label{e1a}\Box_{g'} u'+m^2 u'=0\ ,\quad\Box_{g'}=g'{}^{-1}\nabla'{}^2$$ (here $\nabla'$ is the Levi-Civita connection associated to $g'$)? Is it faster than $O(f(|t|))$ for $g'$ "generic" but $g$ not so?
In other words, which is the best "structurally stable" global time decay rate we can expect from smooth solutions of the Klein-Gordon and wave equation with compactly supported initial data if $(M,g)$ has non-compact Cauchy hypersurfaces (to prevent situations where we know for sure there is no decay at all)?
I expect that loss of decay due to trapping should "generically" disappear but loss of decay due to caustics (which is supposedly milder) should "generically" remain, but I am not sure if this intuition is correct.
(Remark: Müller and Sánchez (Lorentzian Manifolds Isometrically Embeddable in $\mathbb{L}^N$, Trans. Amer. Math. Soc. 363 (2011) 5367-5379, arXiv:0812.4439 [math.DG]) and Minguzzi (On the Existence of Smooth Cauchy Steep Time Functions, Class. Quantum Grav. 33 (2016) 115001, arXiv:1601.05932 [gr-qc]) have shown that any globally hyperbolic $(M,g)$ admits a steep Cauchy time function)
