On the exponential and polynomial convergence for a delayed wave equation without displacement

Abstract

This article places primary emphasis on improving the asymptotic behavior of a multi-dimensional delayed wave equation in the absence of any displacement term. In the first instance, the delay is assumed to occur in the boundary. Then, invoking Bardos–Lebeau–Rauch (BLR) geometric condition (Bardos et al., 1992; Lebeau and Robbiano, 1997) on the domain, the exponential convergence of solutions to their equilibrium state is proved. In turn, an internal delayed wave equation is considered in the second instance, where the three-dimensional domain possesses trapped ray and hence the (BLR) geometric condition (Bardos et al., 1992, Lebeau and Robbiano, 1997) does not hold. Moreover, the internal damping is localized. In such a situation, polynomial convergence results are established. These two findings improve earlier results of Ammari and Chentouf (2017), Phung (2017) and Stahn (2017).