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Semilinear wave
equations on asymptotically de Sitter, Kerr–de Sitter and
Minkowski spacetimes
Peter Hintz and András Vasy |
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Vol. 8 (2015), No. 8, 1807–1890
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Abstract |
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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 nontrapping 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 -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. |
Keywords
semilinear waves, asymptotically de Sitter spaces, Kerr–de
Sitter space, Lorentzian scattering metrics,
b-pseudodifferential operators, resonances, asymptotic
expansion
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Mathematical Subject Classification 2010
Primary: 35L71
Secondary: 35L05, 35P25
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Milestones
Received: 27 November 2013
Revised: 10 April 2015
Accepted: 3 September 2015
Published: 23 December 2015
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Authors |
| Peter Hintz | |
| Department of Mathematics University of California, Berkeley 970 Evans Hall #3840 Berkeley, CA 94720-3840 United States |
Department of Mathematics Stanford University 450 Serra Mall, Bldg. 380 Stanford, CA 94305-2125 United States |
| András Vasy | |
| Department of Mathematics Stanford University 450 Serra Mall, Bldg. 380 Stanford, CA 94305-2125 United States |
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