The semilinear Klein-Gordon equation in de Sitter spacetime

TL;DR

This paper investigates the blow-up behavior and global existence of solutions to the semilinear Klein-Gordon equation in de Sitter spacetime, revealing conditions under which solutions blow up or exist globally based on energy and parameters.

Contribution

It provides new results on blow-up phenomena and global existence thresholds for the semilinear Klein-Gordon equation in de Sitter space, extending previous understanding.

Findings

Large energy solutions blow up for all p>1

Small energy solutions have a critical p(m,n) for global existence

Representation formulas and Kato's lemma are used for analysis

Abstract

In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation $\Box_g \phi-m^2 \phi = -|\phi |^p $ with the small mass $m \le n/2$ in de Sitter space-time with the metric $g$. We prove that for every $p>1$ the large energy solution blows up, while for the small energy solutions we give a borderline $p=p(m,n)$ for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of the Kato's lemma.