# Asymptotic behavior of a delayed wave equation without displacement term

## TL;DR

This paper analyzes the long-term behavior of a delayed wave equation lacking a displacement term, proving well-posedness and establishing convergence rates without geometric control conditions.

## Contribution

It demonstrates the asymptotic stability and logarithmic convergence of solutions without requiring geometric conditions like the BLR condition.

## Key findings

- Solutions converge to stationary states depending on initial data
- Logarithmic decay rate established without geometric control conditions
- Well-posedness proven using semigroup theory

## Abstract

This paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any position 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 in the control zone, the logarithmic convergence is proved by using an interpolation inequality combined with a resolvent method.