Wave equations with strong damping in Hilbert spaces

We study the Cauchy problem for abstract dissipative equations in Hilbert spaces generalizing wave equations with strong damping terms in RN or exterior domains. Our main result is a generalized diffusion phenomenon: the long time asymptotics of strongly damped wave equations is a combination of solutions of diffusion and wave equations. In particular, we obtain sharp decay estimates. The proofs rely on the energy method in the Fourier space and its generalization based on the spectral theorem for self-adjoint operators.