The semilinear Klein-Gordon equation in de Sitter spacetime

In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation ${\mathrm{\square }}_g\varphi -m^2\varphi =-\mathrm{\mid }\varphi {\mathrm{\mid }}^p$ with the small mass $m\le n\mathrm{/}2$ 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.