  We study asymptotics for solutions of Maxwell's equations, in fact of the
Hodge-de Rham equation $(d+\delta)u=0$ without restriction on the form degree,
on a geometric class of stationary spacetimes with a warped product type
structure (without any symmetry assumptions), which in particular include
Schwarzschild-de Sitter spaces of all spacetime dimensions $n\geq 4$. We prove
that solutions decay exponentially to $0$ or to stationary states in every form
degree, and give an interpretation of the stationary states in terms of
cohomological information of the spacetime. We also study the wave equation on
differential forms and in particular prove analogous results on
Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our
analysis and deduce asymptotics and decay for solutions of Maxwell's equations,
the Hodge-de Rham equation and the wave equation on differential forms on
Kerr-de Sitter spacetimes with small angular momentum.
