  We study in special limiting cases quasinormal modes of massive scalar fields
in the Schwarzschild-de Sitter black hole backgrounds. We determine the lower
limit on the mass parameter of the scalar field that allows the waves with
quasinormal frequencies to propagate to infinity, showing that it depends on
the spacetime parameters only. Then we discuss in the large multipole number
limit quasinormal modes, whose frequencies can be directly related to the
unstable circular photon geodesics. In the large scalar mass approximation, we
demonstrate the new interesting phenomenon of slowly decaying resonances, that
are strongly related to the maximum of the effective potential of the massive
scalar field, which is located at the static radius of the Schwarzschild-de
Sitter spacetimes, where the cosmic repulsion is just balanced by the black
hole attraction.
