Note on exponential and polynomial convergence for a delayed wave equation without displacement

TL;DR

This paper investigates the asymptotic behavior of a multi-dimensional delayed wave equation without displacement, demonstrating exponential convergence under boundary delay with geometric conditions and polynomial convergence in domains with trapped rays, improving prior results.

Contribution

It provides new convergence results for delayed wave equations, establishing exponential decay under boundary delay with geometric conditions and polynomial decay in trapped ray domains, extending previous work.

Findings

Exponential convergence for boundary delayed wave equations under BLR condition.

Polynomial convergence for internal delayed wave equations in trapped ray domains.

Improved theoretical understanding of asymptotic behavior in delayed wave equations.

Abstract

This note 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 a geometric condition BLR on the domain, the exponential convergence of solutions to their equilibrium state is proved. The strategy adopted of the proof is based on an interpolation inequality combined with a resolvent method. In turn, an internal delayed wave equation is considered in the second case, where the domain possesses trapped ray and hence (BLR) geometric condition does not hold. In such a situation polynomial convergence results are established. These finding improve earlier results of Ammari-Chentouf and Phung.