We consider the initial-value problem of the abstract wave equation with dissipation whose coefficient tends to $0$ as $t \to \infty$. In the case that the coefficient of the dissipation is a positive constant, Ikehata-Nishihara and Chill-Haraux obtained the decay estimate of the difference between the solution of this equation and the solution of the corresponding abstract heat equation. The purpose of this paper is to show the decay estimate of the difference between the solution of the above equation and the solution of the corresponding abstract parabolic equation. We also show the decay estimate for the solution of this dissipative wave equation.
"Asymptotic behavior for abstract wave equations with decaying dissipation." Adv. Differential Equations 11 (4) 419 - 456, 2006.