On the asymptotic stability of wave equations coupled by velocities of anti-symmetric type

TL;DR

This paper investigates the asymptotic stability of coupled wave equations with anti-symmetric velocity coupling, demonstrating logarithmic stability under specific geometric conditions using frequency domain methods.

Contribution

It establishes the necessity of the intersection condition between coupling and damping domains for stability in 1-D wave systems.

Findings

System is logarithmically stable with smooth initial data.

Intersection of coupling and damping domains is necessary for stability.

Provides an example confirming the geometric assumption's necessity.

Abstract

In this paper, we study the asymptotic stability of two wave equations coupled by velocities of anti-symmetric type via only one damping. We adopt the frequency domain method to prove that the system with smooth initial data is logarithmically stable, provided that the coupling domain and the damping domain intersect each other. Moreover, we show, by an example, that this geometric assumption of the intersection is necessary for 1-D case.