  In this paper we study the Cauchy problem for semi-linear de Sitter models
with power non-linearity. The model of interest is \[ \phi_{tt} - e^{-2t}
\Delta \phi + n\phi_t+m^2\phi=|\phi|^p,\quad
(\phi(0,x),\phi_t(0,x))=(f(x),g(x)),\] where $m^2$ is a non-negative constant.
We study the global (in time) existence of small data solutions. In particular,
we show the interplay between the power $p$, admissible data spaces and
admissible spaces of solutions (in weak sense, in sense of energy solutions or
in classical sense).
