In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove L p − L q estimates for the solutions of the equation with and without a source term.

We examine the solutions of the semilinear wave equation, and, in particular, of the ϕ p model of quantum field theory in the curved space-time. More exactly, for 1 < p < 4 we prove that solution of the massless self-interacting scalar field equation in the Einstein-de Sitter universe has finite lifespan.

a b s t r a c tWe consider waves, which obey the semilinear Klein-Gordon equation, propagating in the Friedmann-Lemaître-Robertson-Walker spacetimes. The equations in the de Sitter and Einstein-de Sitter spacetimes are the important particular cases. We show the global in time existence in the energy class of solutions of the Cauchy problem.

We consider the wave propagating in the Einstein and de Sitter space-time. The covariant d’Alembert’s operator in the Einstein and de Sitter space-time belongs to the family of the non-Fuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the Lp−Lq estimates for solutions.

We prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The coefficients of the equation depend on spatial variables as well, that make results applicable to the space-time with the time slices being Riemannian manifolds. IntroductionIn this paper we prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The coefficients of the equation depend on spatial variables as well, that make results applicable to the space-time with the time slices being Riemannian manifolds. In the spatially flat de Sitter model, these slices are R 3 , while in the spatially closed and spatially open cases these slices can be the three-sphere S 3 and the three-hyperboloid H 3 , respectively (see, e.g., [12, p.113]).The metric g in the de Sitter space-time is defined as follows, g 00 = g 00 = −1, g 0j = g 0j = 0, g ij (x, t) = e 2t σ ij (x), i, j = 1, 2, . . . , n, where n j=1 σ ij (x)σ jk (x) = δ ik , and δ ij is Kronecker's delta. The metric σ ij (x) describes the time slices. In the quantum field theory the matter fields are described by a function ψ that must satisfy equations of motion. In the case of a massive scalar field, the equation of motion is the semilinear Klein-Gordon equation generated by the metric g:Here m is a physical mass of the particle. In physical terms this equation describes a local selfinteraction for a scalar particle. A typical example of a potential function would be V (ψ) = ψ 4 . The semilinear equations are also commonly used models for general nonlinear problems.The covariant Klein-Gordon equation in the de Sitter space-time in the coordinates is