Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes

Abstract

In this paper we show the small data solvability of suitable semilinear wave and Klein-Gordon equations on geometric classes of spaces, which include so-called asymptotically de Sitter and Kerr-de Sitter spaces, as well as asymptotically Minkowski spaces. These spaces allow general infinities, called conformal infinity in the asymptotically de Sitter setting; the Minkowski type setting is that of non-trapping Lorentzian scattering metrics introduced by Baskin, Vasy and Wunsch. Our results are obtained by showing the global Fredholm property, and indeed invertibility, of the underlying linear operator on suitable L^2-based function spaces, which also possess appropriate algebra or more complicated multiplicative properties. The linear framework is based on the b-analysis, in the sense of Melrose, introduced in this context by Vasy to describe the asymptotic behavior of solutions of linear equations. An interesting feature of the analysis is that resonances, namely poles of the inverse of the Mellin transformed b-normal operator, which are `quantum' (not purely symbolic) objects, play an important role.

81 pages, 6 figures. v2: The appendix of v1 has been removed and posted independently on arxiv. The introduction has been rewritten, connecting text added throughout the paper, and typos corrected. One of the hypotheses of Theorem 5.2 has been removed. v3: the published version, with typos corrected and references updated, includes a brief review of b-analysis in section 2

References in corpus (3)

• Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole
• Resonance projectors and asymptotics for r-normally hyperbolic trapped sets
• Non-trapping estimates near normally hyperbolic trapping

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• Resonances for obstacles in hyperbolic space
• Microlocal analysis near null infinity in asymptotically flat spacetimes
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