We consider the initial-value problem of abstract wave equations with weak dissipation. We show that under conditions on the dissipation coefficient and its derivative the solutions to the abstract dissipative equation are closely related to solutions of the free problem multiplied by a decay function. This paper gives the counterpart to a recent paper of T. Yamazaki [Adv. Differential Equ., 11(4):419--456, 2006], where effective dissipation terms and the relation to the corresponding abstract parabolic problem are considered.
"Scattering and modified scattering for abstract wave equations with time-dependent dissipation." Adv. Differential Equations 12 (10) 1115 - 1133, 2007.