A N-dimensional elastic\viscoelastic transmission problem with Kelvin-Voigt damping and non smooth coefficient at the interface

TL;DR

This paper studies the stabilization and energy decay of a multidimensional coupled wave system with Kelvin-Voigt damping, employing spectral analysis, Carleman estimates, and frequency domain techniques to establish stability results under various geometric conditions.

Contribution

It provides new stability and decay rate results for a complex N-dimensional elastic/viscoelastic system with non-smooth interface coefficients, without requiring geometric control conditions.

Findings

Strong stability without geometric conditions

Non-uniform stability demonstrated via spectral analysis

Polynomial decay rates established under specific geometric configurations

Abstract

We investigate the stabilization of a multidimensional system of coupled wave equations with only one Kelvin Voigt damping. Using a unique continuation result based on a Carleman estimate and a general criteria of Arendt Batty, we prove the strong stability of the system in the absence of the compactness of the resolvent without any geometric condition. Then, using a spectral analysis, we prove the non uniform stability of the system. Further, using frequency domain approach combined with a multiplier technique, we establish some polynomial stability results by considering different geometric conditions on the coupling and damping domains. In addition, we establish two polynomial energy decay rates of the system on a square domain where the damping and the coupling are localized in a vertical strip.