A general decay result for a memory-type thermoelasticity with second sound

In this paper, we consider an n-dimensional system of thermoelasticity with second sound in the presence of a viscoelastic term acting in the domain. We prove a general decay result from which the usual exponential and polynomial decay results are only special cases.     

A stability result of a Timoshenko system with a delay term in the internal feedback

In this paper, we consider a Timoshenko system with a delay term in the feedback and prove a stability result. The beam is clamped at the endpoints and has, in addition to an internal damping, a feedback with a delay.Under an appropriate assumption on the weights of the two feedbacks, we prove the well-posedness of the system and establish an exponential decay result for the case of equal-speed wave propagation.     

关于连接梁的最优衰减率问题

研究具有耗散结点的连接梁的最优指数衰减率问题,该系统由于能量的衰减而导致弯矩在结点处间断,我们的方法是证明系统的一组广义征元生成状态空间的Riesz基,从而证明最优指数衰减率可由系统的谱确定。     

On the rate of decay in interacting continua with memory

This paper is concerning the linear theory of isothermal interacting continua with memory. We consider anti-plane shear deformations in a mixture of two elastic solids where the dissipation mechanisms can be the viscosity in one of the components and the viscosity with respect to the relative displacement. We have seen that when the only dissipation mechanism applies on the relative displacement we cannot expect the exponential decay for the solutions. We have also analyzed the case when the viscosity mechanism applies on a constituent. We have seen that generically the decay is of exponential type. However if the coupling constitutive parameter vanishes the decay is slow.     

A general decay result of a viscoelastic equation with infinite history and nonlinear damping

In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and in the presence of an infinite-memory term. We prove an explicit and general decay result using some properties of the convex functions. Our approach allows a wider class of kernels, from which those of exponential decay type are only special cases.     

GENERAL DECAY FOR A DIFFERENTIAL INCLUSION OF KIRCHHOFF TYPE WITH A MEMORY CONDITION AT THE BOUNDARY

In this article, we consider a differential inclusion of Kirchhoff type with a memory condition at the boundary. We prove the asymptotic behavior of the corresponding solutions. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.     

General decay for a differential inclusion of Kirchhoff type with a memory condition at the boundary

In this article, we consider a differential inclusion of Kirchhoff type with a memory condition at the boundary. We prove the asymptotic behavior of the corresponding solutions. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.     

Energy decay of variable-coefficient wave equation with acoustic boundary conditions and delay

In this paper, we consider a variable-coefficient wave equation with memory type acoustic boundary conditions and a constant time delay in the boundary feedback. Using the Riemannian geometry method, we prove the exponential decay of the system with memory type acoustic boundary conditions and a constant time delay under some suitable assumptions.     

A new stability number of the Bresse‐Cattaneo system

In this paper, we consider the Bresse‐Cattaneo system with a frictional damping term and prove some optimal decay results for the L²‐norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results     

A general stability result for a nonlinear wave equation with infinite memory

In this work we consider a nonlinear wave problem in the presence of an infinite-memory term and prove an explicit and general stability result. Our approach allows a wider class of kernels, among which those of exponential decay type, usually considered in the literature, are only special cases.     

Energy decay for Porous-thermo-elasticity systems of memory type

Linear systems of porous-thermo-elasticity including a memory term in one dimension are studied. We establish an exponential and polynomial decay results.     

Exponential decay for a viscoelastically damped timoshenko beam

Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-known that the system is exponentially stable if the kernel in the memory term is sub-exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions: What if the kernel is tested against a different function (say Gamma) other than the exponential function? Would there still be stability? In the affirmative, what kind of decay rate we get? It is proved that for a non-decreasing function “Gamma” whose “logarithmic derivative” is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.     

Exponential decay in one-dimensional Type II/III thermoelasticity with two porosities

In this paper, we consider the theory of thermoelasticity with a double porosity structure in the context of the Green–Naghdi Types II and III heat conduction models. For the Type II, the problem is given by four hyperbolic equations, and it is conservative (there is no energy dissipation). We introduce in the system a couple of dissipation mechanisms in order to obtain the exponential decay of the solutions. To be precise, we introduce a pair of the following damping mechanisms: viscoelasticity, viscoporosities, and thermal dissipation. We prove that the system is exponentially stable in three different scenarios: viscoporosity in one structure jointly with thermal dissipation, viscoporosity in each structure, and viscoporosity in one structure jointly with viscoelasticity. However, if viscoelasticity and thermal dissipation are considered together, undamped solutions can be obtained     

Existence and uniform decay for Euler–Bernoulli beam equation with memory term

In this article we prove the existence of the solution to the mixed problem for Euler–Bernoulli beam equation with memory term. The existence is proved by means of the Faedo–Galerkin method and the exponential decay is obtained by making use of the multiplier technique combined with integral inequalities due to Komornik. Copyright © 2004 John Wiley & Sons, Ltd.     

General energy decay in a Timoshenko-type system of thermoelasticity of type III with a viscoelastic damping

In this paper we consider a one-dimensional linear thermoelastic system of Timoshenko type, where the heat conduction is given by Green and Naghdi theories. We prove a general decay result, from which the exponential and polynomial decays are only special cases.     

Stability results for laminated beam with thermo-visco-elastic effects and localized nonlinear damping

In this article, we investigate a one-dimensional thermoelastic laminated beam system with nonlinear damping and viscoelastic dissipation on the effective rotation angle and through heat conduction in the interfacial slip equations. Under minimal conditions on the relaxation function and the relationship between the coefficients of the wave propagation speed of the first two equations, we show that the solution energy has an explicit and optimal decay rate from which the exponential and polynomial stability are just particular cases. Moreover, we establish a weaker decay result in the case of non-equal wave of speed propagation and give some examples illustrate our results. This work extends and improves the earlier results in the literature, particularly the result of Mukiawa et al. (2021).     

Existence and decay rates for a coupled Balakrishnan-Taylor viscoelastic system with dynamic boundary conditions

In this paper, we are concerned with a coupled viscoelastic wave system with Balakrishnan-Taylor dampings, dynamic boundary conditions, source terms, and past histories. Under suitable assumptions on relaxation functions and source terms, we prove the global existence of solutions by potential well theory and we establish a more general decay result of energy, in which the exponential decay and polynomial decay are only special cases, by introducing suitable energy and perturbed Lyapunov functionals.     

Stability of Timoshenko systems with past history

We consider vibrating systems of Timoshenko type with past history acting only in one equation. We show that the dissipation given by the history term is strong enough to produce exponential stability if and only if the equations have the same wave speeds. Otherwise the corresponding system does not decay exponentially as time goes to infinity. In the case that the wave speeds of the equations are different, which is more realistic from the physical point of view, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.     

On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping

This paper is concerned with the asymptotic behavior of solutions of the critical generalized Korteweg-de Vries equation in a bounded interval with a localized damping term. Combining multiplier techniques and compactness arguments it is shown that the problem of exponential decay of the energy is reduced to prove the unique continuation property of weak solutions. A locally uniform stabilization result is derived.

     

Exponential decay for a viscoelastic problem with a singular kernel

In this paper we consider a problem which arises in viscoelasticity. We prove exponential decay of solutions for the problem with a memory term involving a kernel which is singular at zero. This is established by introducing an appropriate Lyapunov type functional and using the energy method. This work extends earlier results.