In this paper we describe the behavior of solutions of the Klein–Gordon equation,
, on Lorentzian
manifolds
that are anti de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian analogues
of the so-called Riemannian conformally compact (or asymptotically hyperbolic)
spaces, in the sense that the metric is conformal to a smooth Lorentzian metric
on
, where
has a nontrivial boundary,
in the sense that
,
with
a boundary defining function. The boundary is conformally timelike for
these spaces, unlike asymptotically de Sitter spaces studied before by Vasy
and Baskin, which are similar but with the boundary being conformally
spacelike.
Here we show local well-posedness for the Klein–Gordon equation, and also global
well-posedness under global assumptions on the (null)bicharacteristic flow, for
below the
Breitenlohner–Freedman bound,
.
These have been known before under additional assumptions. Further, we describe
the propagation of singularities of solutions and obtain the asymptotic behavior (at
)
of regular solutions. We also define the scattering operator, which in this
case is an analogue of the hyperbolic Dirichlet-to-Neumann map. Thus, it is
shown that below the Breitenlohner–Freedman bound, the Klein–Gordon
equation behaves much like it would for the conformally related metric,
, with
Dirichlet boundary conditions, for which propagation of singularities was shown by
Melrose, Sjöstrand and Taylor, though the precise form of the asymptotics is
different.
Keywords
asymptotics, wave equation, anti de Sitter space,
propagation of singularities
