Stabilization Rates for the Damped Wave Equation with Hölder-Regular Damping

Abstract

This paper studies decay rates for the energy of solutions of the damped wave equation on the torus. It considers dampings invariant in one direction and equal to a sum of squares of nonnegative functions with some number of derivatives. If such a damping vanishes only on a small enough strip then the energy decays at rate 1/t 3/4 .The proof uses a positive commutator argument and relies on a pseudodifferential calculus for low regularity symbols.

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…3 ) [8,137], no matter how large the support of b. The fact that the regularity of b may play a more important role than its support is also nicely illustrated in [52,87,88,97].…”

Sharp polynomial decay rates for the damped wave equation with Hölder-like damping

“…(1) Our result is especially interesting when W = D near the set where |x| = σ. Then, by Theorem 1.1 of [Kle19], the value of α in (5) is the best possible. (2) As is clear from the reduction to (8) at the beginning of the proof below, the same proof gives the same result (with the same constant C) if the torus T = (R/2πZ) x × (R/2πZ) y is replaced by another product (R/2πZ) x × Σ y , where Σ is any compact Riemannian manifold.…”

“…In [Kle19], the second author shows that, if W = D near σ, then (3) holds with α = (β+2)/(β+4). In the case of constant damping on a strip (W = D and β = 0) the result that (3) holds with α = 2/3 is due to Stahn [Sta17], and the result that it does not hold for α > 2/3 is due to Nonnenmacher [AL14].…”

“…We consider dampings invariant in one direction and bounded above and below by multiples of x β near the boundary of the support and show decay at rate 1/t β+2 β+3 . In the case where W vanishes exactly like x β this result is optimal by [Kle19]. The proof uses a version of the Morawetz multiplier method.…”

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

“…with damping b(x, y) ≥ 0, and initial data (u 0 , v 0 ) lying in some sufficiently regular Sobolev space precised below. Our main goal is to study how the geometry of the damping region ω := (x, y) ∈ T 2 : b(x, y) > 0 and the regularity of b affect the energy decay rate of the solutions to (1.2), and what is the influence of the subellipticity stemmed from the Baouendi-Grushin operator in this decay rate, in comparison with the elliptic Laplacian ∆ on the torus (see [2], [17], [22], [28]).…”