Polynomial and non exponential stability of a weak dissipative Bresse system

TL;DR

This paper investigates the stability of a Bresse system with linear frictional damping, showing it does not stabilize exponentially but converges polynomially, using semigroup, energy, and frequency domain methods.

Contribution

It provides the first proof that linear frictional damping does not lead to exponential stability in the Bresse system and establishes polynomial decay rates.

Findings

Linear frictional damping does not ensure exponential stability.

Solutions decay polynomially over time.

Well-posedness is established via semigroup theory.

Abstract

In this paper, we study the Bresse system in a bounded domain with linear frictional dissipation working only on the veridical displacement. The longitudinal and shear angle displacements are free. Our first main result is to prove that, independently from the velocities of waves propagations, this linear frictional dissipation does not stabilize exponentially the whole Bresse system. Our second main result is to show that the solution converges to zero at least polynomially. The proof of the well-posedness of our system is based on the semigroup theory. The stability results will be proved using a combination of the energy method and the frequency domain approach.