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We consider the massive scalar field equation $\Box_{g_{RN}} \phi = m^2 \phi$on any subextremal Reissner--Nordstr\"{o}m exterior metric $g_{RN}$. We provethat solutions with localized initial data decay pointwise-in-time at thepolynomial rate $t^{-\frac{5}{6}+\delta}$ in any spatially compact region(including the event horizon), for some small $ \delta\leq \frac{1}{23} $.Moreover, assuming the validity of the Exponent Pair Conjecture on exponentialsums in Number Theory, our result implies that decay upper bounds hold at therate $t^{-\frac{5}{6}+\epsilon}$, for any arbitrarily small $\epsilon>0$.  In our previous work, we proved that each fixed angular mode decays at theexact rate $t^{-\frac{5}{6}}$, thus the upper bound $t^{-\frac{5}{6}+\epsilon}$is sharp, up to a $t^{\epsilon}$ loss. Without the restriction to a fixedangular mode, the solution turns out to have an unbounded Fourier transform dueto discrete frequencies associated to quasimodes, and caused by the occurrenceof stable timelike trapping. Our analysis nonetheless shows thatinverse-polynomial asymptotics in $t$ still hold after summing over all angularmodes.

Polynomial time decay for solutions of the Klein--Gordon equation on a subextremal Reissner--Nordstr\"{o}m black hole