The semilinear Klein-Gordon equation in de Sitter spacetime

Abstract

In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation ${\square }_g\varphi -m^2\varphi =-|\varphi {|}^p$ 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.

References in corpus (1)

• The wave equation on Schwarzschild-de Sitter spacetimes