The Generalized Diffusion Phenomenon and Applications

Abstract

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.

Remarks on the asymptotic behavior of the solution to damped wave equations

“…For the damping satisfying a(x) ∼ (1 + |x|) −1 , Ikehata, Todorova and Yordanov [12] proved the energy decay. For abstract setting, the diffusion phenomena was proved in [21] but their results and the proof seem to slightly different from ours.…”

Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

“…After that, in our previous results [45,46], we extended the result of [52] to exterior domains and more general (but bounded) damping satisfying the condition (1.2) with α ∈ (−1, 0]. Radu, Todorova and Yordanov [41] and Nishiyama [36] proved the diffusion phenomena in an abstract setting. However, for the damping term increasing at the spatial infinity, there is no result about the diffusion phenomena, while Khader [16,17] studied energy estimates and global existence of small solutions for some nonlinear problems.…”

Local energy decay and diffusive phenomenon in a dissipative wave guide

“…For a slowly decaying absorption index (apxq " x ´ρ with ρ Ps0, 1s), we refer to [ITY13,Wak14] (recall that if the absorption index is of short range (ρ ą 1), then we recover the properties of the undamped wave equation, see [BR14,Roy]). Finally, results on an abstract setting can be found in [CH04,RTY10,Nis,RTY16].…”