Wave equation with hyperbolic boundary condition: a frequency domain approach

TL;DR

This paper analyzes the stability and decay rates of a wave equation with a hyperbolic boundary condition involving a coupled wave equation and dissipative feedback, using a frequency domain approach.

Contribution

It introduces a novel frequency domain method to prove semi-uniform stability and polynomial decay for a wave equation with complex boundary conditions.

Findings

The closed-loop system generates a semi-uniformly stable semigroup.

Polynomial decay rate established under geometrical conditions.

Resolvent operator growth estimates on the imaginary axis.

Abstract

In this paper, we investigate the stability of the linear wave equation where one part of the boundary, which is seen as a lower-dimensional Riemannian manifold, is governed by a coupled wave equation, while the other part is subject to a dissipative Robin velocity feedback. We prove that the closed-loop equations generate a semi-uniformly stable semigroup of linear contractions on a suitable energy space. Furthermore, under multiplier-related geometrical conditions, we establish a polynomial decay rate for strong solutions. This is achieved by estimating the growth of the resolvent operator on the imaginary axis.