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

TL;DR

This paper investigates the decay rates of wave energy in cylindrical waveguides with boundary damping, revealing sharp polynomial decay rates depending on the geometry and damping conditions.

Contribution

It establishes sharp decay rates for wave energy in cylindrical waveguides with boundary damping, including cases where geometric control conditions are not satisfied.

Findings

Sharp $t^{-1/2}$ decay in product cylinders with boundary damping.

$t^{-1/3}$ decay in non-product cylinders with boundary damping.

Decay rates depend on the geometry and damping bounds.

Abstract

We study the decay of global energy for the wave equation with H\"older continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact 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.