Theoretical and numerical study of the decay in a viscoelastic Bresse System

TL;DR

This paper investigates the energy decay in a one-dimensional viscoelastic Bresse system, providing theoretical decay results and demonstrating optimal decay rates for specific conditions and relaxation functions.

Contribution

It offers new theoretical decay estimates for the Bresse system with finite memory, including optimal decay rates for polynomially decaying relaxation functions.

Findings

Proved general decay results for the system's energy.

Established conditions for equal and non-equal wave speeds.

Showed optimal decay rates matching relaxation function decay.

Abstract

In this paper, we consider a one-dimensional finite-memory Bresse system with homogeneous Dirichlet-Neumann-Neumann boundary conditions. We prove some general decay results for the energy associated with the system in the case of equal and non-equal speeds of wave propagation under appropriate conditions on the relaxation function. In addition, we show by giving an example that in the case of equal speeds of wave propagation and for certain polynomially decaying relaxation functions, our result gives an optimal decay rate in the sense that the decay rate of the system is exactly the same as that of the relaxation function considered.