Energy decay for linear dissipative wave equations in exterior domains

Abstract

In earlier works [2,5], we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turns out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation.

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).…”

Local energy decay and diffusive phenomenon in a dissipative wave guide

“…See [Nis03,MN03,HO04,Nar04]. See also [Ike02,AIK] for the damped wave equation on an exterior domain. 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]).…”

Energy decay in a wave guide with dissipation at infinity

“…In [Mat76], [TY09] or [AIK15], where global energy decay is studied, the initial data is localized (compactly supported, or at least in L 2 ∩ L q for some q ∈ [1, 2[). Theorem 1.1 also contains this kind of result if we take δ 1 = 0 and δ 2 > 0.…”