Polynomial stabilization of non-smooth direct/indirect elastic/viscoelastic damping problem involving Bresse system

TL;DR

This paper investigates the polynomial stability of a coupled elastic/viscoelastic Bresse system with various damping configurations, establishing energy decay rates and generalizing previous results in the field.

Contribution

It introduces new multiplier techniques and broadens the understanding of stabilization for complex coupled wave systems with Kelvin-Voigt damping.

Findings

Different energy decay rates depending on damping configuration

Generalization of previous stability results for Bresse and Timoshenko systems

Conditions under which polynomial stability is achieved

Abstract

We consider an elastic/viscoelastic transmission problem for the Bresse system with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish different types of energy decay rate (see the table of stability results below). Our results generalize and improve many earlier ones in the literature and in particular some studies done on the Timoshenko system with Kelvin-Voigt damping.