Global solutions for semilinear Klein–Gordon equations in FLRW spacetimes

We 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.