The Generalized Diffusion Phenomenon and Applications

We study the asymptotic behavior of solutions to dissipative wave equations involving two non-commuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place, as solutions of such equations approach solutions of diffusion equations at large times. When the diffusion semigroup has the Markov property and satisfies a Nash-type inequality, we obtain precise estimates for the consecutive diffusion approximations and remainder. We present several important applications including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first decay estimates for nonlocal wave equations with damping; the decay rates are sharp.