On the asymptotic behavior of the quasi-static problem for a linear viscoelastic fluid

In this paper we study the quasi-static problem for a viscoelastic fluid by means of the concept of minimal state. This implies the use of a different free energy defined in a wider space of data. The existence and uniqueness is proved in this new space and the asymptotic decay for the problem with non vanishing supplies is obtained for a large class of memory kernels, including those presenting an exponential or polynomial decay.     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

A general method for proving sharp energy decay rates for memory-dissipative evolution equations

This Note is concerned with stabilization of hyperbolic systems by a distributed memory feedback. We present here a general method which gives energy decay rates in terms of the asymptotic behavior of the kernel at infinity. This method, which allows us to recover in a natural way the known cases (exponential, polynomial, …), applies to a large quasi-optimal class of kernels. It also provides sharp energy decay rates compared to the ones that are available in the literature. We give a general condition under which the energy of solutions is shown to decay at least as fast as the kernel at infinity. To cite this article: F. Alabau-Boussouira, P. Cannarsa, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type

In this paper we consider linear porous-thermoelasticity systems, in a bounded domain, where the memory-type damping is acting on a part of the boundary. We establish a general decay result, for which the usual exponential and polynomial decay rates are just special cases. Our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and, therefore, generalizes and improves on earlier results from the literature.     

General decay of solutions of a wave equation with a boundary control of memory type

In this paper we consider a semilinear wave equation, in a bounded domain, where the memory-type damping is acting on a part of the boundary. We establish a general decay result, from which the usual exponential and polynomial decay rates are only special cases. Our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and, therefore, generalizes and improves earlier results in the literature.     

On a large class of kernels yielding exponential stability in viscoelasticity

In this paper we prove exponential decay of solutions for a problem which appears in viscoelasticity. The conditions on the admissible kernels are relaxed so as to allow for more kernels to be treated. Namely, the smallness of the kernels is replaced by the smallness of the set where the kernel is flat. This work extends previous works and improves in particular a recent result by Pata [16]. This is established by introducing two lemmas, an idea due to Pata and using the energy method.     

Exponential stability of a linear viscoelastic bar with thermal memory

In this paper we study a one-dimensional evolution problem arising in the theory of linear thermoviscoelasticity with hereditary heat conduction. Depending on the istantaneous conductivity K₀, both Coleman-Gurtin (K₀>0) and Gurtin-Pipkin (K₀=0) heat flow theories are involved. In any case, the exponential stability of the corresponding semigroup is proved for a class of memory functions including weakly singular kernels. In order to achieve the exponential decay of the energy, we assume that mechanical and thermal memory kernels decay exponentially for large time. Entrata in Redazione il 23 luglio 1998.     

Sharp adaptation for spherical inverse problems with applications to medical imaging

This paper examines the estimation of an indirect signal embedded in white noise for the spherical case. It is found that the sharp minimax bound is determined by the degree to which the indirect signal is embedded in the linear operator. Thus, when the linear operator has polynomial decay, recovery of the signal is polynomial, whereas if the linear operator has exponential decay, recovery of the signal is logarithmic. The constants are determined for these classes as well. Adaptive sharp estimation is also carried out. In the polynomial case a blockwise shrinkage estimator is needed while in the exponential case, a straight projection estimator will suffice. The framework of this paper include applications to medical imaging, in particular, to cone beam image reconstruction and to diffusion magnetic resonance imaging. Discussion of these applications are included.     

Polynomial contractions and Lyapunov regularity

The purpose of this note is twofold: to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time, and to show that it persists under sufficiently small linear and nonlinear perturbations. The notion of polynomial contraction mimics the notion of exponential contraction, but with the exponential decay replaced by a polynomial decay. We show that this behavior is exhibited by a large class of dynamics, by giving necessary conditions in terms of “polynomial” Lyapunov exponents. Finally, we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations. We also consider the case of nonuniform polynomial contractions, for which the Lyapunov stability is not uniform.     

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators. We prove sub-exponential decay for functions in Gevrey classes and exponential decay for real analytic functions.

     

Asymptotic stability of semigroups associated with linear weak dissipative systems with memory

We study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations. We show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution. In the case of the lack of exponential decay, we find the polynomial rate of decay of the solution. Some examples are given.     

Boundary stabilization of solutions of a nonlinear system of Timoshenko type

In this paper we are concerned with a multidimensional Timoshenko system subjected to boundary conditions of memory type. We establish general rate decay results. The usual exponential and polynomial decay rates are only special cases.     

General decay of solutions in one-dimensional porous-elastic system with memory

In this work we consider a one-dimensional porous-elastic system with memory effects. It is well-known that porous-elastic system with a single dissipation mechanism lacks exponential decay. In contrary, we prove that the unique dissipation given by the memory term is strong enough to exponentially stabilize the system, depending on the kernel of the memory term and the wave speeds of the system. In fact, we prove a general decay result, for which exponential and polynomial decay results are special cases. Our result is new and improves previous results in the literature.     

Integro-differential equations of hyperbolic type with positive definite kernels

The main purpose of this work is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in Hilbert spaces. For kernels that decay exponentially at infinity and possess strongly positive definite primitives, the exponential stability of weak solutions is obtained in the energy norm. It is also shown that this theory applies to several examples of kernels with possibly variable sign, and to a problem in nonlinear viscoelasticity.     

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.     

Damping of end effects in a thermoelastic theory

The aim of this paper is to present a spatial decay estimate in the thermoelasticity of Type III. We prove that the rate of decay is bounded below by an exponential of a second degree polynomial of the distance.     

Stability of solution for a mixture of thermoelastic of type III

In this work, we analyze the existence, uniqueness, and asymptotic behavior of solution to the model of a thermoelastic mixture of type III. We establish sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

General boundary stabilization of memory type in thermoelasticity of type III

In this paper, we consider a multi-dimensional system of thermoelasticity type III with a viscoelastic damping acting on a part of the boundary. We establish a general decay result, from which the usual exponential and polynomial decay rates are only special cases.     

Stabilization of a viscoelastic Timoshenko beam

A viscoelastic Timoshenko beam is investigated. We prove an exponential decay of solutions for a large class of kernels with weaker conditions than the existing ones in the literature. This will allow the use of other types of viscoelastic material for Timoshenko type beams than the usually used ones.