具有局部非线性反馈的非均质Timoshenko梁的能量衰减估计

该文考虑具有局部非线性反馈的非均质Timoshenko梁的能量衰减估计.由非线性算子半群理论得到系统的适定性;应用乘子方法,给出了系统的能量衰减估计.     

Stability to weakly dissipative Timoshenko systems

In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.     

The global classical solution to compressible system with fractional viscous term

We consider the initial value problem to the fractional system of motions for compressible viscous fluids in this paper. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the large-time behavior of the solution, where solutions converge to a constant steady state exponentially in time.     

一类弹性梁的能量衰减估计

研究一类弹性梁的耗散性.首先应用非线性算子半群理论证明了系统的适定性;进而运用能量方法结合乘子技巧得到弹性梁系统的能量衰减估计.     

Exponential Decay Rate for a Timoshenko Beam with Boundary Damping

The exponential decay rate of a Timoshenko beam system with boundary damping is studied. By asymptotically analyzing the characteristic determinant of the system, we prove that the Timoshenko beam system is a Riesz system; hence, its decay rate is determined via its spectrum. As a consequence, by showing that the imaginary axis neither has an eigenvalue on it nor is an asymptote of the spectrum, we conclude that the system is exponentially stable.     

Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III

This paper is concerned with the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III. We first obtain the well-posedness of the system by using semigroup method. We then investigate the asymptotic behaviour of the system through the perturbed energy method. We prove that the energy of system decays exponentially in the case of equal wave speeds and decays polynomially in the case of nonequal wave speeds. Under the case of nonequal wave speeds, we also investigate the lack of exponential stability of the system.     

Sine-Gordon方程的动态边界反馈镇定

考虑一端固定,控制输入在另一端的Sine-Gordon方程的动态边界反馈镇定问题.我们给出闭环系统的适定性,并利用乘子方法得到在动态边界反馈控制下闭环系统的多项式衰减率.     

具有Kelvin-Voigt阻尼的弱耦合系统的能量衰减估计

研究具有 Kelvin-Voigt 阻尼的弱耦合系统。首先在合适的假设条件下,应用线性算子半群理论证明了系统的适定性;进而运用线性算子半群的频域定理证明了具有Kelvin-Voigt阻尼的弱耦合梁―弦系统的能量是一致指数衰减的。     

Global Solution and Exponential Stability for a Laminated Beam with Fourier Thermal Law

This paper focuses on the long-time dynamics of a thermoelastic laminated beam modeled from the well-established Timoshenko theory. From mathematical point of view, the study system consists of three hyperbolic motion equations coupled with the parabolic equation governed by Fouriers law of heat conduction and, in consequence, does not belong to one of the classical categories of PDE. We have proved the well-posedness and exponential stability of the system. The well-posedness is given by Hille-Yosida theorem. For the exponential decay we applied the energy method by introducing a Lyapunov functional.     

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.     

Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas

This paper is concerned with the global well-posedness and exponential stability of solutions to a one-dimensional model for the viscous radiative and reactive gas with higher-order kinetics. We prove that under rather general assumptions on the heat conductivity κ, for any large smooth initial data, the problem admits a unique global classical solution. Moreover, the solution will exponentially decay to the unique steady state as time goes to infinity.     

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     

Exponential and polynomial decay for a laminated beam with Fourier's law of heat conduction and possible absence of structural damping

We study the well-posedness and decay properties of a onedimensional thermoelastic laminated beam system either with or without structural damping, of which the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is wellposed by using the Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method. When the speeds are not equal, whether the system without structural damping may has polynomial stability is left as an open problem.     

The Navier–Stokes problem modified by an absorption term

It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|^σ?2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 < σ < 2 and exponentially decay in time if σ = 2. In the special case of a suitable force field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided 1 < σ < 2. We also prove that for non-zero body forces decaying at a power-time rate, the solutions decay at analogous power-time rates if σ > 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.     

On global classical solutions of the three dimensional non-relativistic Vlasov–Darwin System

We are concerned with the global well-posedness of the non-relativistic Vlasov–Darwin system with generalized variables approach in three dimensions. We obtain the global existence and uniqueness of classical solutions for the perturbation of global solutions with specified decay conditions. And generalizing the result of the quasi-spherical-symmetry case, we prove the existence and uniqueness of the global classical solution of the system when initial data sufficiently closes to a fixed spherically symmetric function. Moreover, we obtain asymptotic behavior for the Darwin potentials in both cases.     

Stability and optimality of decay rate for a weakly dissipative Bresse system

This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Exponential decay domain of energy for wave equation under feedback control

1.IntroductionLetnbeaboundedopendomaininH"(n22)withsmoothboundaryr,u~u(s)betheouternormalofrat8.GivenavectorxoEE",denotem(x)=x--xo(xER"),R=sup{im(x)IIxEfi}.SetWeconsiderthefollowingwaveequstiononnwithboundaryfeedback:whereK=(m'u)k(8),L=f(m'u),with(,n...     

Decay estimates for second order evolution equations with memory

This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t→+∞. Our approach is based on integral inequalities and multiplier techniques.These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system.     

A 2-D free boundary problem with onset of a phase and singular elliptic boundary value problems

Numerous models of industrial processes, such as diffusion in glassy polymers or solidification phenomena, lead to general one phase free boundary value problems with phase onset.The classical well-posedness of a fast diffusion approximation to the concerned free boundary value problems is proved. The analysis is performed via a singular change of variables leading to a singular system in a fixed domain. An existence and regularity theory for classical solutions is developed for the relevant underlying class of singular elliptic boundary value problems and is then used to prove the well-posedness for the models considered in which these are coupled to Hamilton-Jacobi or to parabolic evolution equations.