  We study the global dynamics of the wave equation, Maxwell's equation and the
linearized Bianchi equations on a fixed anti-de Sitter (AdS) background.
Provided dissipative boundary conditions are imposed on the dynamical fields we
prove uniform boundedness of the natural energy as well as both degenerate
(near the AdS boundary) and non-degenerate integrated decay estimates.
Remarkably, the non-degenerate estimates "lose a derivative". We relate this
loss to a trapping phenomenon near the AdS boundary, which itself originates
from the properties of (approximately) gliding rays near the boundary. Using
the Gaussian beam approximation we prove that non-degenerate energy decay
without loss of derivatives does not hold. As a consequence of the
non-degenerate integrated decay estimates, we also obtain pointwise-in-time
decay estimates for the energy. Our paper provides the key estimates for a
proof of the non-linear stability of the anti-de Sitter spacetime under
dissipative boundary conditions. Finally, we contrast our results with the case
of reflecting boundary conditions.
