Asymptotic behavior of a delayed wave equation without displacement term

This paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any displacement term. First, it is shown that the problem is well-posed in the sense of semigroups theory. Thereafter, LaSalle’s invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. More importantly, without any geometric condition such as BLR condition (Bardos et al. in SIAM J Control Optim 30:1024–1064, 1992; Lebeau and Robbiano in Duke Math J 86:465–491, 1997) in the control zone, the logarithmic convergence is proved by using an interpolation inequality combined with a resolvent method.