Vol. 8, No. 8, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes

Peter Hintz and András Vasy

Vol. 8 (2015), No. 8, 1807–1890
Abstract

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 L2-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
Mathematical Subject Classification 2010
Primary: 35L71
Secondary: 35L05, 35P25
Milestones
Received: 27 November 2013
Revised: 10 April 2015
Accepted: 3 September 2015
Published: 23 December 2015
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