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

TL;DR

This paper proves the small data solvability of semilinear wave and Klein-Gordon equations on various asymptotically de Sitter, Kerr-de Sitter, and Minkowski spacetimes by establishing the invertibility of associated linear operators using advanced microlocal analysis techniques.

Contribution

It introduces a unified linear framework based on b-analysis to analyze semilinear equations on complex geometric backgrounds, including new results on invertibility and resonance analysis.

Findings

Proved global Fredholm property and invertibility of linear operators.

Established small data solvability for semilinear wave and Klein-Gordon equations.

Analyzed the role of resonances in the solution behavior.

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.