  In this note we provide some precise estimates explaining the diffusive
structure of partially dissipative systems with time-dependent coefficients
satisfying a uniform Kalman rank condition. Precisely, we show that under
certain (natural) conditions solutions to a partially dissipative hyperbolic
system are asymptotically equivalent to solutions of a corresponding parabolic
equation.
  The approach is based on an elliptic WKB analysis for small frequencies in
combination with exponential stability for large frequencies due to results of
Beauchard and Zuazua and arguments of perturbation theory.
