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We describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an $${\varepsilon}$$ derivative loss in the angular directions.

Decay and Non-Decay of the Local Energy for the Wave Equation on the De Sitter–Schwarzschild Metric