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

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 $t^{-1\mathrm{/}2}$-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 $t^{-1\mathrm{/}3}$-decay when the damping is uniformly bounded from below on the cylindrical wall.