A classification for wave models with time-dependent mass and speed of propagation

In this paper, we study the long time behavior of energy solutions for a class of wave equation with time-dependent mass and speed of pro\-pagation. We introduce a classification of the potential term, which clarifies whether the solution behaves like the solution to the wave equation or Klein-Gordon equation. Moreover, $L^q-L^2, q\in [1, 2]$ estimates for scale-invariant models are derived and applied to obtain global in time small data energy solutions for the semilinear Klein-Gordon equation in anti de Sitter spacetime.