Numerical Exponential Decay to Dissipative Bresse System

We consider the Bresse system with frictional dissipative terms acting in all the equations. We show the exponential decay of the solution by using a method developed by Z. Liu and S. Zheng and their collaborators in past years. The numerical computations were made by using the finite difference method to prove the theoretical results. In particular, the finite difference method in our case is locking free

Non-equilibrium thermodynamics of damped Timoshenko and damped Bresse systems

In this paper, we cast damped Timoshenko and damped Bresse systems into a general framework for non-equilibrium thermodynamics, namely the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework. The main ingredients of GENERIC consist of five building blocks: a state space, a Poisson operator, a dissipative operator, an energy functional, and an entropy functional. The GENERIC formulation of damped Timoshenko and damped Bresse systems brings several benefits. First, it provides alternative ways to derive thermodynamically consistent models of these systems by construct- ing building blocks instead of invoking conservation laws and constitutive relations. Second, it reveals clear physical and geometrical structures of these systems, e.g., the role of the energy and the entropy as the driving forces for the reversible and irreversible dynamics respectively. Third, it allows us to introduce a new GENERIC model for damped Timoshenko systems that is not existing in the literature.Comment: 22 pages, revise

The lack of exponential stability of a Bresse system subjected only to two dampings

In this paper, we study the indirect boundary stabilization of a Bresse system with only two dissipation laws. This system, which models the dynamics of a beam, is a hyperbolic system with three wave speeds. We study the asymptotic behaviour of the eigenvalues and of the eigenvectors of the underlying operator in the case of three distinct wave velocities which is not physically relevant. Since the imaginary axis is proved to be an asymptote for one family of eigenvalues, the stability can not be exponential. Of course, this paper is only interesting from a mathematical point of view

ENERGY DECAY RATES FOR THE BRESSE-CATTANEO SYSTEM WITH WEAK NONLINEAR BOUNDARY DISSIPATION

In this paper, we consider a one-dimensional Bresse system with Cattaneo’s type heat conduction and a nonlinear weakly dissipative boundary feedback localized on a part of the boundary. We show the well-posedness, using the semigroup theory, and establish an explicit and general decay rate result without imposing a specific growth assumption on the behavior of damping terms near zero

Well-Posedness and Polynomial energy decay rate of a transmission problem for Rayleigh beam model with heat conduction

In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem's well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt-Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams