Stability and instability results for coupled waves with delay term

Abstract

In this paper, we consider a coupled system of two wave equations. One of these equations is conservative, and the other has damping and delay terms. If the damping acts with more force than the delay term, we show polynomial stability for strong solutions of the system. Explicit decay rates are found, and the optimality of those rates is discussed. On the other hand, if the damping acts with the same or less force than the delay term, then we obtain a result of instability by constructing a sequence of time delays and initial data such that the solutions are not asymptotically stable.