Optimal stabilization rate for the wave equation with hyperbolic boundary condition

TL;DR

This paper proves that the energy of solutions to the wave equation with hyperbolic boundary conditions decays at an optimal rate of 1/t, under certain geometric conditions, using resolvent estimates and high-frequency analysis.

Contribution

It establishes the sharp decay rate for wave equations with hyperbolic boundary conditions, extending previous results to mixed boundary conditions and providing resolvent estimates.

Findings

Energy decay rate is 1/t for solutions.

Decay rate is proven to be sharp.

Results apply to mixed boundary conditions with geometric control.

Abstract

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.