The semilinear Klein-Gordon equation in de Sitter spacetime

Abstract

In this article we study the blow-up phenomena for the solutions of the semilinearKlein-Gordon equation $\square_g$ $\phi-m^2 \phi = -|\phi |^p $ with the small mass $m \le n/2$ in de Sitter spacetime with the metric $g$.We prove that for every $p>1$ large energy solutions blow 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 andon some generalizations of Kato's lemma.