  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.
