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

## TL;DR

This paper investigates the global existence of small data solutions to semi-linear de Sitter models with power non-linearity, analyzing how the non-linearity power and data spaces influence solution behavior.

## Contribution

It provides new results on the conditions for global existence of solutions in various function spaces for semi-linear de Sitter models.

## Key findings

- Global existence depends on the power p and initial data spaces.
- Different solution concepts (weak, energy, classical) are considered.
- Interplay between non-linearity and spacetime geometry is characterized.

## Abstract

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).