M.L. Santos

Exponential stability for the Timoshenko system with two weak dampings

In this paper we consider a linear system of Timoshenko type beam equations with frictional dissipative terms. 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. This method is very different from some others in the literature, such as the traditional energy method. It is our hope that the reader will find the method presented in this work is powerful and simple.

Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary

Abstract We consider a nonlinear wave equation of Kirchhoff type with memory condition at the boundary and we study the asymptotic behavior of the corresponding solutions. We proved that the energy decay with the same rate of decay of the relaxation function, that is, the energy decays exponentially when the relaxation function decay exponentially and polynomially when the relaxation function decay polynomially.

A boundary condition with memory for Kirchhoff plates equations

In this paper, we study the stability of solutions for Kirchhoff plates equations with a memory condition working at the boundary. We show that such dissipation is strong enough to produce exponential decay to the solution, provided the relaxation functions also decays exponentially. When the relaxation functions decays polynomially, we show that the solution decays polynomially and with the same rate.

Asymptotic behavior to Bresse system with past history

In this paper we consider the Bresse system with past history acting in the shear angle displacement. We show the exponential decay of the solution if and only if the wave speeds are the same. On the contrary, we show that the Bresse system is polynomial stable with optimal decay rate. The systems of equations considered here introduce new mathematical difficulties in order to determine the asymptotic behavior. As far as the authors know, there have been no contributions made in this sense.

Polynomial Stability to Three-Dimensional Magnetoelastic Waves

In this work we show that the energy associated to the linear three-dimensional magneto-elastic system decays polynomially to zero as time goes to infinity, provided the initial data is smooth enough.In this work we show that the energy associated to the linear three-dimensional magneto-elastic system decays polynomially to zero as time goes to infinity, provided the initial data is smooth enough.

Solution and Asymptotic Behaviour for a Nonlocal Coupled System of Reaction-Diffusion

This paper concerns with the existence, uniqueness and asymptotic behaviour of the solutions for a nonlocal coupled system of reaction-diffusion. We prove the existence and uniqueness of weak solutions by the Faedo-Galerkin method and exponential decay of solutions by the classic energy method. We improve the results obtained by Chipot-Lovato and Menezes for coupled systems. A numerical scheme is presented.

Energy decay in piezoelectric systems

We establish that when a dissipation term is introduced in the equilibrium equations of a piezoelectric body, the energy decays exponentially and we give an estimate of the decay rate.

On the Kirchhoff plates equations with thermal effects and memory boundary conditions

In this paper we study the existence of weak and strong global solutions and uniform decay of the energy to the Kirchhoff plates equations with thermal effect and memory conditions working at the boundary. We show that the dissipation produced by the memory effect not depend on the present values of temperature gradient. That is, we show that the dissipation produced by memory effect is strong enough to produce exponential decay of the solution provided the relaxation functions also decays exponentially. When the relaxation functions decays polynomially, we show that the solution decays polynomially with the same rate.

Stability to the dissipative Reissner–Mindlin–Timoshenko acting on displacement equation

In this paper, we show that there exists a critical number that stabilises the Reissner–Mindlin–Timoshenko system with frictional dissipation acting only on the equation for the transverse displacement. We identify that the Reissner–Mindlin–Timoshenko system has two speed characteristics v 1 2 := K /ρ 1 and v 2 2 := D /ρ 2 and we show that the system is exponentially stable if only if \begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*} In the general case, we prove that the system is polynomially stable with optimal decay rate. Numerical experiments using finite differences are given to confirm our analytical results. Our numerical results are qualitatively in agreement with the corresponding results from dynamical in infinite dimensional.

Uniform Decay Rates of Solutions to a Nonlinear Wave Equation with Boundary Condition of Memory Type

In this article we study the hyperbolic problem (1) where Ω is a bounded region in R n whose boundary is partitioned into disjoint sets Γ0, Γ1. We prove that the dissipation given by the memory term is strong enough to assure exponential (or polynomial) decay provided the relaxation function also decays exponentially (or polynomially). In both cases the solution decays with the same rate of the relaxation function.