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.     

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.     

The exponential decay of a system of nonlinear wave equations

The exponential decay of a system of nonlinear wave equations with initial boundary values is considered. We have some sufficient conditions that ensure that the energy admits exponential decay by a compactness uniqueness argument and the energy estimates.     

Energy decay for Timoshenko systems of memory type

Linear systems of Timoshenko type equations for beams including a memory term are studied. The exponential decay is proved for exponential kernels, while polynomial kernels are shown to lead to a polynomial decay. The optimality of the results is also investigated.     

Comparative exponential dichotomies and column diagonal dominance

It is shown that the dichotomy obtained by Lazer for diagonally dominant linear systems is a weaker condition than that of exponential dichotomy but that the two conditions are equivalent in the case of bounded coefficients. Thus Fink's result that exponential decay of all solutions implies exponential dichotomy does not extend to the case of mixed growth and decay. It is also shown that column dominant systems of mixed sign admit exponential dichotomies when the coefficients are bounded.     

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 exact rates of decay of solutions of linear systems of Volterra equations with delay

This paper considers the resolvent of a finite-dimensional linear convolution Volterra integral equation. The main results give conditions which ensure that the exact rate of decay of the resolvent can be determined using a positive weight function related to the kernel. The decay rates can be exponential or subexponential. Many other related results on exact rates of exponential and subexponential decay of solutions of Volterra integro-differential equations are given. We also present an application to a linear compartmental system with discrete and continuous lags.     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L _p spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge in weighted L _p spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying mask, we find a characterization for the corresponding refinable function to lie in weighted L ₂ spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay, which can be viewed as an extension of the Butterworth filters.     

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincaré maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.     

A note on fractional moments for the one-dimensional continuum Anderson model

We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one-dimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any single-site distribution with bounded, compactly supported density.     

On the exponential decay of the Euler–Bernoulli beam with boundary energy dissipation

We study the asymptotic behavior of the Euler–Bernoulli beam which is clamped at one end and free at the other end. We apply a boundary control with memory at the free end of the beam and prove that the “exponential decay” of the memory kernel is a necessary and sufficient condition for the exponential decay of the energy.     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.Received: January 13, 2004     

Decroissance exponentielle du noyau de la chaleur sur la diagonale (I)

Summary We give examples based upon large deviation's theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal. Using Malliavin calculus, we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal. We give lower bound in some particular case by using the Bismut's condition.     

Delay-range-dependent exponential stability criteria and decay estimation for switched Hopfield neural networks of neutral type

This paper is concerned with the problem of delay-range-dependent global exponential stability and decay estimation for a class of switched Hopfield neural networks (SHNNs) of neutral type. An average dwell time method is introduced into switched Hopfield neural networks. By constructing a new Lyapunov–Krasovskii functional and designing a switching law, some criteria are proposed for guaranteeing exponential stability for a given system, while the exponential decay estimation is explicitly developed for the states. A numerical example is provided to demonstrate the effectiveness of the main results.     

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.     

记忆型梁方程强解的全局吸引子

记忆型梁方程出现于—般的Kirchhoff粘弹性梁模型中．本文在记忆核满足指数衰退的条件下证明了系统的能量也是指数衰退的．进一步,通过对条件(C)的验证获得了系统强解的全局吸引子．     

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.

     

Decay rates of energy of the 1D damped original nonlinear wave equation

We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity.