On the stability of the Bresse system with frictional damping

TL;DR

This paper investigates the stability of the Bresse system with frictional damping, revealing conditions under which the system stabilizes or remains unstable, and introduces a new stability number based on system parameters.

Contribution

It demonstrates that damping on the third equation stabilizes the entire Bresse system and introduces a novel stability number depending on system parameters.

Findings

Damping on the second equation does not lead to decay.

Damping on the third equation stabilizes the system.

A new stability number is derived for the system.

Abstract

In this paper, we consider the Bresse system with frictional damping terms and prove some optimal decay results for the $L^2$-norm of the solution and its higher order derivatives. In fact, if we consider just one damping term acting on the second equation of the solution, we show that the solution does not decay at all. On the other hand, by considering one damping term alone acting on the third equation, we show that this damping term is strong enough to stabilize the whole system. In this case, we found a completely new stability number that depends on the parameters in the system. In addition, we prove the optimality of the results by using eigenvalues expansions. Our obtained results have been proved under some assumptions on the wave speeds of the three equations in the Bresse system.