Stability results of coupled wave models with locally memory in a past history framework via non smooth coefficients on the interface

TL;DR

This paper studies the stability of coupled wave equations with local viscoelastic damping and past history effects, establishing conditions for exponential or polynomial decay based on wave speeds.

Contribution

It provides new stability criteria for coupled wave models with non smooth interface coefficients and past history damping, including conditions for exponential and polynomial decay.

Findings

Strong stability proven via Arendt-Batty criterion.

Exponential stability occurs when wave speeds are equal.

Polynomial decay rate of 1/t when wave speeds differ.

Abstract

In this paper, we investigate the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non smooth coefficients. First, using a general criteria of Arendt-Batty, we prove the strong stability of our system. Second, using a frequency domain approach combined with the multiplier method, we establish the exponential stability of the solution if and only if the two waves have the same speed of propagation. In case of different speed propagation, we prove that the energy of our system decays polynomially with rate 1/t. Finally, we show the lack of exponential stability if the speeds of wave propagation are different.