Stability results of a distributed problem involving Bresse system with history and/or Cattaneo law under fully Dirichlet or mixed boundary conditions

On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction

We investigate the stability of three thermoelastic beam systems with hyperbolic heat conduction. First, we study the Bresse-Gurtin-Pipkin system, providing a necessary and sufficient condition for the exponential stability and the optimal polynomial decay rate when the condition is violated. Second, we obtain analogous results for the Bresse-Maxwell-Cattaneo system, completing an analysis recently initiated in the literature. Finally, we consider the Timoshenko-Gurtin-Pipkin system and we find the optimal polynomial decay rate when the known exponential stability condition does not hold. As a…

具有全齐次Dirichlet边界条件的热弹性Timoshenko系统与Bresse系统缺乏指数稳定性

Lack of exponential stability for thermoelastic Timoshenko and Bresse systems with full homogeneous Dirichlet boundary conditions

本文针对热弹性Timoshenko与Bresse系统，证明其在全齐次Dirichlet边界条件且不满足等波速时无指数稳定性，结合逆结论完成系数相关的指数稳定性刻画，证明引入边界修正项满足边界条件。

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

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…

On the stability of Bresse system with one discontinuous local internal Kelvin-Voigt damping on the axial force

In this paper, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin-Voigt type acting on the axial force, under fully Dirichlet boundary conditions. First, using a general criteria of Arendt-Batty, we prove the strong stability of our system. Finally, using a frequency domain approach combined with the multiplier method, we prove that the energy of our system decays polynomially with different rates.

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

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 th…

The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization

In this paper, we investigate the stabilization of a linear Bresse system with one singular local frictional damping acting in the longitudinal displacement, under fully Dirichlet boundary conditions. First, we prove the strong stability of our system. Next, using a frequency domain approach combined with the multiplier method, we establish the exponential stability of the solution if and only if the three waves have the same speed of propagation. On the contrary, we prove that the energy of our system decays polynomially with rates $t^{-1}$ or $t^{-\frac{1}{2}}$.