Sharp Polynomial Decay for Waves Damped from the Boundary in Cylindrical Waveguides

Abstract

We study the decay of global energy for the wave equation with Hölder continuous damping placed on the C 1,1 -boundary of compact and noncompact waveguides with star-shaped cross-sections. We show there is sharp t −1/2 -decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is t −1/3 -decay when the damping is uniformly bounded from below on the cylindrical wall.

Stabilisation of Waves on Product Manifolds by Boundary Strips

“…There have been some previous results on the decay of waves in cylindrical domains when the damping is uniformly positive on Γ: the t −δ -decay for some δ > 0 was shown in [Phu08], and the t −1/3 -decay could be shown using the method in [Nis13]. In [Wan21] we showed the sharp t −1/2 -decay when X is star-shaped. Theorem 1.3 removes such star-shaped assumption.…”

“…The t −1/2 -decay given in Theorem 1.3 is more optimal on product manifolds. See also the references within [Wan21].…”