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

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Abstract

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.

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, UMR 5251 du CNRS, Université de Bordeaux I, 351 cours de la Libération, 33 405, Talence cedex, France

    Jean-François Bony & Dietrich Häfner

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  1. Jean-François Bony
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  2. Dietrich Häfner
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Correspondence to Jean-François Bony.

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Communicated by G.W. Gibbons

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Bony, JF., Häfner, D. Decay and Non-Decay of the Local Energy for the Wave Equation on the De Sitter–Schwarzschild Metric. Commun. Math. Phys. 282, 697–719 (2008). https://doi.org/10.1007/s00220-008-0553-y

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  • DOI: https://doi.org/10.1007/s00220-008-0553-y

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