Wave Equation with Hyperbolic Boundary Condition: a Frequency Domain Approach

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