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

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.