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The Cauchy problem for the Klein-Gordon equation under the quartic potentialis considered in the de Sitter spacetime. The existence of the global solutionis shown based on the mechanism of the spontaneous symmetry breaking for thesmall positive Hubble constant. The effects of the spatial expansion andcontraction on the problem are considered.

The Cauchy problem for the Klein-Gordon equation under the quartic potential in the de Sitter spacetime