  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}
