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If $(M,g)=(\mathbb{R}^d,\eta)$ 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 $$\tag{3}\label{e3}\|u(x^0,\cdot)\|_{L^\infty(\mathbb{R}^{d-1})}=O\Big(|x^0|^{-\frac{d-2}{2}}\Big)\ .$$

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 Eisntein 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}:

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 $$\|u\|_{L^\infty(\Sigma_t)}=O(f(|t|))$$ 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?

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}:

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?

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 $$\|u\|_{L^\infty(\Sigma_t)}=O(f(|t|))$$ 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?

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 $$\|u\|_{L^\infty(\Sigma_t)}=O(f(|t|))$$ 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?