In this paper, we study the long time behavior of energy solutions for a class of wave equation with time-dependent potential and speed of propagation. 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, the derived linear estimates are applied to obtain global (in time) small data energy solutions for the Cauchy problem to semilinear Klein-Gordon models with power nonlinearity.
"A classification for wave models with time-dependent potential and speed of propagation." Adv. Differential Equations 23 (11/12) 847 - 888, November/December 2018.