Global well-posedness of quasilinear wave equations on asymptotically de Sitter spaces

Abstract

We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity $L^2$-based function spaces and using iterative arguments for the non-linear problems. The linear analysis is accomplished in two parts: Firstly, a regularity theory is developed by means of a calculus for pseudodifferential operators with non-smooth coefficients, similar to the one developed by Beals and Reed, on manifolds with boundary. Secondly, the asymptotic behavior of solutions to linear equations is studied using standard b-analysis, introduced in this context by Vasy; in particular, resonances play an important role.