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

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