Semilinear Hyperbolic Equations in Curved Spacetime

This is a survey of the author's recent work rather than a broad survey of the literature. The survey is concerned with the global in time solutions of the Cauchy problem for matter waves propagating in the curved spacetimes, which can be, in particular, modeled by cosmological models. We examine the global in time solutions of some class of semililear hyperbolic equations, such as the Klein-Gordon equation, which includes the Higgs boson equation in the Minkowski spacetime, de Sitter spacetime, and Einstein & de Sitter spacetime. The crucial tool for the obtaining those results is a new approach suggested by the author based on the integral transform with the kernel containing the hypergeometric function.\\ {\bf Mathematics Subject Classification (2010):} Primary 35L71, 35L53; Secondary 81T20, 35C15.\\ {\bf Keywords:} \small {de Sitter spacetime; Klein-Gordon equation; Global solutions; Huygens' principle; Higuchi bound}