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

In this paper we consider the Cauchy problem for semi-linear de Sitter models with power non-linearity. The model of interest is ϕtt−e−2tΔϕ+nϕt+m2ϕ=|ϕ|p,(ϕ(0,x),ϕt(0,x))=(f(x),g(x)),where m2 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).