|We consider the asymptotic behavior of solutions to dissipative wave equations
involving two non-commuting self-adjoint operators in Hilbert space.
The main result is that the abstract diffusion phenomenon takes place.
We obtain precise estimates for consecutive diffusion approximations and
remainder. In this process we use an abstract version of weighted estimates.
When the diffusion semigroup has the Markov property, relying on
the maximal regularity, we get sharp results. We apply the abstract results
to derive optimal decay estimates for dissipative hyperbolic equations with
variable coefficients in an exterior domain. Applications to nonlocal settings
provide new decay rates for solutions of nonlocal dissipative wave equations.