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We study the asymptotic behavior of solutions to dissipative wave equations involving two noncommuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place. Thus 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 remainders. 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. [ABSTRACT FROM AUTHOR]

Global well-posedness in H²(R3)×H¹(R³) is shown for nonlinear wave equations of the form ...u + f(u) + g(ut) = 0, where t ∈ R+. The main assumption is that the nonlinear damping g(ut) behaves like |ut|m-1ut with m ≥ 2 and the defocusing nonlinearity f(u) is like |u|p-1u with p ≥ 2. The result also applies to certain exponential functions, such as f(u) = sinhu. It is observed that the nonlinear damping gives rise to a new monotone quantity involving the second-order derivatives of u and leading to a priori estimates for initial data of any size. Global well-posedness in H¹(R³) × L²(R³) is shown for the same equation in the critical case f(u) = u5 and g(ut) = |ut|2/3ut. The main tool is a new estimate for the solution of the nonlinear equation in L4(R+, L12(R³)). [ABSTRACT FROM AUTHOR]

Abstract: We establish weighted -estimates for the wave equation with variable damping in , where with and . In particular, we show that the energy of solutions decays at a polynomial rate if for large . We derive these results by strengthening significantly the multiplier method. This approach can be adapted to other hyperbolic equations with damping. [Copyright &y& Elsevier]