  We prove an asymptotic stability result for a linear coupled
hyperbolic-elliptic system on a large class of singular background spacetimes
in CMC gauge on the n-torus. At each spatial point these background spacetimes
are perturbations of Kasner-like solutions of the Einstein-scalar field
equations which are not required to be close to the homogeneous and isotropic
case. We establish the existence of a homeomorphism between Cauchy data for
this system and a set of functions naturally associated with the asymptotics in
the contracting direction, which we refer to as asymptotic data. This yields a
complete characterization of the degrees of freedom of all solutions of this
system in terms of their asymptotics. Spatial derivative terms can in general
not be fully neglected which yields a clarification of the notion of asymptotic
velocity term dominance (AVTD).

General Relativity and Quantum Cosmology;Analysis of PDEs