We consider the charged Klein-Gordon equation outside the De Sitter-Reissner-Nordström black hole. The product of the charge of the field and the charge of the black hole is the coupling term and plays an important role in the study of the equation. The natural conserved quantities associated to solutions are not positive because of the coupling and energies can grow in time: this is superradiance.
We show that when the charge product is sufficiently small with respect to the mass of the field, the local energy decays in time; this result is obtained by means of a resonance expansion of the local propagator. The decay is exponential if we allow a polynomial loss of derivatives in the angular directions and is valid through the event and cosmological horizons.
The decay of the local energy allows us to show an asymptotic completeness result. As the coupling comes from the equation itself and not from the geometry (contrary to the De Sitter-Kerr case), we build a Kaluza-Klein extension of the spacetime in order to interpret the scattering as transport onto horizons along principal null geodesics in the extended spacetime. Asymptotic completeness then implies the existence and the invertibility of trace operators in energy spaces, showing that an abstract Goursat problem is well posed in the extension.