Regularity theory and global existence of small data solutions to semi-linear de Sitter models with power non-linearity

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).