Locally internal stability of weakly elastic Bresse system

TL;DR

This paper investigates the stability of the Bresse system with local dissipation, demonstrating exponential stability under certain conditions and establishing a new polynomial decay rate when dissipation is localized.

Contribution

It extends previous work by analyzing local dissipation effects on the Bresse system and improves the decay rate results.

Findings

Exponential stability under equal speed wave propagation.

New polynomial energy decay rate with local dissipation.

Enhanced understanding of local dissipation effects on system stability.

Abstract

In their paper "Stability to weak dissipative bresse system", Alabau et al. studied the exponential and polynomial stability of the Bresse system with one globally distributed dissipation law. Our goal is to extend their results, by taking into consideration the important case when the dissipation law is locally distributed and to improve the polynomial energy decay rate. We then study the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, we show that the system is exponentially stable. On the contrary, we establish a new polynomial energy decay rate.