Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

TL;DR

This paper studies the stabilization of a one-dimensional coupled wave system with localized viscoelastic damping and time delay, proving strong stability and a polynomial decay rate without relying on resolvent compactness.

Contribution

It introduces a novel stability analysis for coupled wave equations with non-smooth damping and time delay, using Arendt-Batty criteria and frequency domain methods.

Findings

System is strongly stable without resolvent compactness.

Energy decays polynomially at rate 1/t.

Stability results apply to systems with localized viscoelastic damping.

Abstract

The purpose of this paper is to investigate the stabilization of a one-dimensional coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combining with a multiplier method, we prove a polynomial energy decay rate of order 1/t.