Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes

The decay of solutions to the Klein–Gordon equation is studied in two expanding cosmological spacetimes, namelythe de Sitter universe in flat Friedmann–Lemaître–Robertson–Walker (FLRW) form andthe cosmological region of the Reissner–Nordström–de Sitter (RNdS) model. Using energy methods, for initial data with finite higher-order energies, decay rates for the solution are obtained. Also, a previously established decay rate of the time derivative of the solution to the wave equation, in an expanding de Sitter universe in flat FLRW form, is improved, proving Rendall’s conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime.