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September  2009, 2(3): 679-696. doi: 10.3934/dcdss.2009.2.679

The semilinear Klein-Gordon equation in de Sitter spacetime

Karen Yagdjian 1,

1. 

Department of Mathematics, University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78541-2999

Received  October 2008 Revised  March 2009 Published  June 2009

In this article we study the blow-up phenomena for the solutions of the semilinear Klein-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 and on some generalizations of Kato's lemma.
Keywords: blow-up., Klein-Gordon equation, de Sitter spacetime.
Mathematics Subject Classification: Primary: 35L70, 35Q75; Secondary: 35Q4.
Citation: Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679
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