Stabilization of coupled wave equations with viscous damping on cylindrical and non-regular domains: Cases without the geometric control condition

TL;DR

This paper studies the stability of coupled wave equations with viscous damping on complex domains, demonstrating polynomial energy decay rates under various damping and propagation speed conditions without geometric control.

Contribution

It provides new results on polynomial energy decay rates for coupled wave systems with viscous damping on irregular domains without geometric control conditions.

Findings

Energy decays as t^{-1/2} when one wave is damped and speeds are equal.

Energy decays as t^{-1/3} when one wave is damped and speeds differ.

Energy decays as t^{-1} when both waves are damped.

Abstract

In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate $t^{-\frac{1}{2}}$ if the two waves have the same speed of propagation, and with rate $t^{-\frac{1}{3}}$ if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order $t^{-1}$.