Global Analysis of Quasilinear Wave Equations on Asymptotically De Sitter Spaces
نویسنده
چکیده
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 L2-based function spaces and using iterative arguments for the nonlinear 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.
منابع مشابه
Global Well-posedness of Quasilinear Wave Equations on Asymptotically De Sitter Spaces
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 L2-based function spaces and using iterative arguments for the nonlinear problems....
متن کاملGlobal Analysis of Quasilinear Wave Equations on Asymptotically Kerr-de Sitter Spaces
We extend the semilinear framework developed by the two authors in [29] and the non-trapping quasilinear theory developed by the first author [27] to solve quasilinear wave equations with normally hyperbolic trapping. The most well-known example that fits into our general framework is wavetype equations on Kerr-de Sitter space. The key advance is an adaptation of the Nash-Moser iteration to our...
متن کاملSemilinear Wave Equations on Asymptotically De Sitter, Kerr-de Sitter and Minkowski Spacetimes
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 ...
متن کاملStrichartz Estimates on Asymptotically De Sitter Spaces
In this article we prove a family of local (in time) weighted Strichartz estimates with derivative losses for the KleinGordon equation on asymptotically de Sitter spaces and provide a heuristic argument for the non-existence of a global dispersive estimate on these spaces. The weights in the estimates depend on the mass parameter and disappear in the “large mass” regime. We also provide an appl...
متن کاملA Parametrix for the Fundamental Solution of the Klein-gordon Equation on Asymptotically De Sitter Spaces
In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of ( − λ)u = f and to obtain a uniform L estimate for a family of bump functions traveling to infinity.
متن کامل