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

@article{Hintz2015SemilinearWE, title={Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes}, author={P. Hintz and A. Vasy}, journal={Analysis \& PDE}, year={2015}, volume={8}, pages={1807-1890} }

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… Expand

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#### References

SHOWING 1-10 OF 201 REFERENCES

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)

- Mathematics, Physics
- 2010

In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under… Expand

The wave equation on asymptotically anti de Sitter spaces

- Physics, Mathematics
- 2009

In this paper we describe the behavior of solutions of the KleinGordon equation, ( g + λ)u = f , on Lorentzian manifolds (X, g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds… Expand

GLOBAL WELL-POSEDNESS OF QUASILINEAR WAVE EQUATIONS ON ASYMPTOTICALLY DE SITTER SPACES

- Mathematics, Physics
- 2014

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.… Expand

The wave equation on asymptotically de Sitter-like spaces

- Mathematics, Physics
- 2007

In this paper we obtain the asymptotic behavior of solutions of the Klein–Gordon equation on Lorentzian manifolds (X○,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues… Expand

Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates

- Mathematics, Physics
- 2011

In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic spaces, which was introduced recently by the author. This new method in particular constructs the… Expand

Global Analysis of Quasilinear Wave Equations on Asymptotically Kerr-de Sitter Spaces

- Mathematics
- 2016

We consider quasilinear wave equations on manifolds for which infinity has a structure generalizing that of Kerr-de Sitter space; in particular the trapped geodesics form a normally hyperbolic… Expand

The wave equation on Schwarzschild-de Sitter spacetimes

- Physics, Mathematics
- 2007

We consider solutions to the linear wave equation $\Box_g\phi=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an… Expand

Global Existence for the Einstein Vacuum Equations in Wave Coordinates

- Mathematics, Physics
- 2005

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the… Expand

Extensions of the Stability Theorem of the Minkowski Space in General Relativity

- Mathematics, Physics
- 2009

This book consists of two independent works: Part I is 'Solutions of the Einstein Vacuum Equations', by Lydia Bieri. Part II is 'Solutions of the Einstein-Maxwell Equations', by Nina Zipser. A famous… Expand

The semilinear Klein-Gordon equation in de Sitter spacetime

- Mathematics, Physics
- 2009

In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation $\Box_g \phi-m^2 \phi = -|\phi |^p $ with the small mass $m \le n/2$ in de Sitter space-time… Expand