Optimal stabilization rate for the wave equation with hyperbolic boundary condition

We show that the energy of classical solutions to the wave equation with hyperbolic boundary condition (i.e., dynamic Wentzell boundary condition) and damping on the boundary decays like 1/t. In fact we allow mixed boundary conditions: a possibly empty, disjoint part of the boundary may be kept at rest provided that the dynamic part satisfies the geometric control condition. We also prove that this decay rate is sharp. Our results follow from resolvent estimates, which we establish by studying high-frequency quasimodes.