  The decay behavior of a massless scalar field in the Schwarzschild-de Sitter
spacetime is well-known to follow an exponential law at asymptotically late
times $t \rightarrow \infty$. In contrast, a massive scalar field in the
asymptotically flat Schwarzschild background exhibits a decay with oscillatory
(sinusoidal) tails enveloped by a power law. We demonstrate that the asymptotic
decay of a massive scalar field in the Schwarzschild-de Sitter spacetime is
exponential. Specifically, if $\mu M \gg 1$, where $\mu$ and $M$ represent the
mass of the field and the black hole, respectively, the exponential decay is
also oscillatory. Conversely, in the regime of small $\mu M$, the decay is
purely exponential without oscillations. This distinction in decay regimes
underscores the fact that, for asymptotically de Sitter spacetimes, a
particular branch of quasinormal modes, instead of a "tail", governs the decay
at asymptotically late times. There are two branches of quasinormal modes for
the Schwarzschild-de Sitter spacetime: the modes of an asymptotically flat
black hole corrected by a non-zero $\Lambda$-term, and the modes of an empty de
Sitter spacetime corrected by the presence of a black hole. We show that the
latter branch is responsible for the asymptotic decay. When $\mu M$ is small,
the modes of pure de Sitter spacetime are purely imaginary (non-oscillatory),
while at intermediate and large $\mu M$ they have both real and imaginary
parts, what produces the two pictures of the asymptotic decay. In addition, we
show that the asymptotic decay of charged and higher dimensional black hole is
also exponential.
