Local energy decay and diffusive phenomenon in a dissipative wave guide

Abstract

We prove the local energy decay for the wave equation in a wave guide with dissipation at the boundary. It appears that for large times the dissipated wave behaves like a solution of a heat equation in the unbounded directions. The proof is based on resolvent estimates. Since the eigenvectors for the transverse operator do not form a Riesz basis, the spectral analysis does not trivially reduce to separate analyses on compact and Euclidean domains.

Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation

“…The same phenomenon has been observed in an exterior domain (see [Ike02] for a constant absorption index and [AIK15] for an absorption index equal to 1 outside some compact) and in a wave guide (see [Roy18a] for a constant dissipation at the boundary and [MR18] for an asymptotically constant absorption index). For a slowly decaying absorption index (a(x) = ⟨x⟩ −ρ with ρ ∈ (0, 1]) we refer to [TY09], [ITY13], [Wak14] (we recall from [Roy18b] that if a(x) ≲ ⟨x⟩ −ρ with ρ > 1 then we recover the behavior of the undamped wave equation).…”

Stabilisation of Waves on Product Manifolds by Boundary Strips

“…See Remark 1.4(3). This is also the only boundary damping result we know when Y is non-compact, along the local energy decay obtained in [Roy18].…”