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

Abstract

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.

Global in time existence of self-interacting scalar field in de Sitter spacetimes

“…(For more references on the asymptotically de Sitter spaces, see the bibliography in [5], [28].) Hintz and Vasy [19] considered the semilinear wave equations of the form…”

Quasilinear waves and trapping: Kerr-de Sitter space

“…The only paper the authors are aware of on non-linear problems in the Kerrde Sitter setting is their earlier paper [29] in which the semilinear Klein-Gordon equation was studied. There is more work on the linear equation on perturbations of de Sitter-Schwarzschild and Kerr-de Sitter spaces: a rather complete analysis of the asymptotic behavior of solutions of the linear wave equation was given in [46], upon which the linear analysis of [30], described here, is ultimately based.…”

“…If the trapping causes further losses of derivatives, one would need q = q(u)! We refer to [29] for more detail.…”

“…In X for Kerr-de Sitter space there is non-trivial dynamics within the radial set (rotating black hole), but the normal dynamics is again source/sink. In both cases, L = SN * (Y ∩ X), where Y is the event horizon of the black hole or the cosmological horizon of the de Sitter end; see [46] and Figure 5 b (M )), the situation is similar, except one has to use Ψ b (M ) to microlocalize; see [46] and especially [29].…”

Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes

“…[DMP1,DMP2,BJ] for generalizations on the allowed classes of spacetimes). Furthermore, propagation estimates in b-Sobolev spaces of variable order were used recently to show a similar result in the case of the wave equation on asymptotically Minkowski spacetimes [VW], drawing on earlier developments by Vasy et al [BVW,HV,Va1,Va2]. The two methods being however currently limited to a special value of the mass parameter, our focus here is instead on the proof of the Hadamard property of the in and out state for the Klein-Gordon operator P = −✷ g + m 2 for any positive mass m, or more generally for P = −✷ g + V with a real-valued potential V ∈ C ∞ (M ) satisfying an asymptotic positivity condition.…”