非对称阻尼系统特征对一阶导数与二阶导数的计算

提出了一种计算非对称阻尼系统特征对一阶、二阶导数的方法.该方法利用阻尼系统的特征向量计算特征对的导数,避免了状态空间中特征向量的使用,节省了计算量,且不要求系统所有特征值的互异性.最后以两个非对称阻尼系统进行数值试验,数值结果表明提出的方法是有效的.     

The attractor for the two-dimensional weakly damped KdV equation in belt field

This paper aims at presenting a proof of the existence of the attractor for the two-dimensional weakly damped KdV equations in belt field. By using the Bourgain spaces and the orthogonal projection, we prove the existence of the attractor for the reduced equation which is obtained from the two-dimensional weakly damped KdV equation in belt field. By using the time estimate of the blow-up of the KdV equation, we get the compact and bounded aborting set and prove the existence of the attractor for the two-dimensional weakly damped KdV equation in belt field.     

Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain

In this paper, we consider a damped Klein-Gordon equation in a noncylindrical domain. This work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped Klein-Gordon equation in a noncylindrical domain.     

Hereditary effects of exponentially damped oscillators with past histories

This paper presents hereditary effects of exponentially damped oscillators with past histories. Unlike the classical viscously damped oscillators, the nonviscously damped ones involve damping forces which depend on time-histories of vibrating motions via convolution integrals. As a result, equations of motion of such systems are a set of coupled second-order Volterra integro-differential equations. In this work, initial value problems for the integro-differential equations are revisited. The initial conditions should contain time-histories of vibrating motions. Then, initialization response of exponentially damped oscillators is obtained. It is used to characterize the hereditary effects on the dynamic response. At last, stability of initialization response is proved from the theoretical viewpoint and verified by numerical simulations. This reveals that the hereditary effects gradually recede with increasing of time.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system

We consider the problem of sharp energy decay rates for nonlinearly damped abstract infinite-dimensional systems. Direct methods for nonlinear stabilization generally rely on multiplier techniques, and thus are valid under restrictive geometric conditions compared to the optimal geometric optics condition of Bardos et al. (1992) [10]. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely an observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine optimal geometric conditions, as for instance that of Bardos et al. (1992) [10] and the optimal-weight convexity method of the first author (Alabau-Boussouira, 2010 [6], Alabau-Boussouira, 2005 [2]) to deduce very simple and quasi-optimal energy decay rates for nonlinearly locally damped systems. We also show that using arguments based on Russell's principle (Russell, 1978 [24]), one can deduce sharp energy decay rates from the exponential stabilization of the linearly damped system. Our results extend to nonlinearly damped systems, those of Haraux (1989) [14] and Ammari and Tucsnak (2001) [9] which concern linearly damped systems.     

Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation

We give necessary conditions for a sequence of complex numbers closed under complex conjugation to be the spectrum of the weakly damped wave operator. These restrictions are a consequence of some new results on spectral sequences for Hermitian quadratic pencils which are based on Weyl's and Mirsky's classical eigenvalue inequalities. In the case of finite-dimensional weakly damped pencils our conditions are both necessary and sufficient. We also obtain some conditions for overdamped pencils of degree m, and show that some of the inequalities that have to be satisfied in the weakly damped case are now reversed.     

The existence of a random attractor for the three dimensional damped Navier–Stokes equations with additive noise

This article is concerned with the asymptotical behavior of solutions for the three-dimensional damped Navier–Stokes equations with additive noise. Due to the shortage of the existence proof of the existence of random absorbing sets in a more regular phase space, we cannot obtain some kind of compactness of the cocycle associated with the three-dimensional damped Navier–Stokes equations with additive noise by the Sobolev compactness embedding theorem. In this paper, we prove the existence of a random attractor for the three-dimensional damped Navier–Stokes equations with additive noise by verifying the pullback flattening property.     

Large-time behavior of solutions to degenerate damped hyperbolic equations

We investigate the asymptotic behavior of solutions to damped hyperbolic equations involving strongly degenerate differential operators. First we establish the existence of a global attractor for the damped hyperbolic equation under consideration. Then we prove the finite dimensionality of the global attractor.     

Multiple doubly periodic solutions of semi-linear damped beam equations

The purpose of this work is to investigate the weakened Ambrosetti–Prodi type multiplicity results for weak doubly periodic solutions of damped beam equations. By using the topological degree theory, the author obtains a result which is similar to the result for damped wave equations in the literature.     

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time‐dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro‐differential equation with hyperbolic memory kernel. Copyright © 2012 John Wiley & Sons, Ltd.     

GENERALIZED RICCATI TRANSFORMATION AND OSCILLATION FOR LINEAR DIFFERENTIAL EQUATIONS WITH DAMPING

Using generalized Riccati transformation, some new oscillation criteria for damped linear differential equations are established. These results improve and generalize some known oscillation criteria due to A.Wintner, I.V.Kamenev for the undamped linear differential equations, and Sobol, J.S.W.Wong for the damped linear differential equations.     

Stabilization of a transmission wave/plate equation

We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchhoff plate equation. We prove an exponential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate.     

The direct updating of damping and gyroscopic matrices

In this paper we develop an efficient numerical method for the finite element model updating of damped gyroscopic systems. This model updating of damped gyroscopic systems is proposed to incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the damping and gyroscopic matrices that closely match the experimental modal data.     

具弱耗散项的非线性Hartree方程

该文讨论了一类在量子理论中有着较多应用的具耗散项的非线性Hartree方程。分别从耗散系数和初值两个方面讨论了解的整体存在性条件。一方面,利用Strichartz估计,得到仅依赖于耗散系数的整体存在性条件。另一方面,也得到了仅依赖于初值大小的整体存在性条件。而且,还得到了一个整体解存在的小初值准则。     

Periodic problem for the nonlinear damped wave equation with convective nonlinearity

We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy-type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions.     

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.     

Bessel transforms and rational extrapolation

Summary A numerical method is developed which handles the Bessel transform of functions having slow rates of decrease. The method replaces the Bessel transform by a related damped transform for which the sinc quadrature rule provides an efficient and accurate approximation. It is then shown that the value of the original Bessel transform can be obtained from the damped transform by extrapolation with the Thiele algorithm.     

一类广义耦合的非线性波动方程组时间周期解的存在性

研究了一类广义耦合的非线性波动方程组关于时间周期解的问题.首先利用Galerkin方法构造近似时间周期解序列,然后利用先验估计和Laray-Schauder不动点原理,证明近似时间周期解序列的收敛性,从而得到该问题时间周期解的存在性.     

A new exact solution of a damped quadratic non-linear oscillator

In this paper, we derive a new exact solution of the damped quadratic nonlinear oscillator (Helmholtz oscillator) based on the developed solution for the undamped case by the Jacobi elliptic functions. It is interesting to see that both of the damped Duffing oscillator and Helmholtz oscillator possess solutions that follow closely to the undamped case, and even the solution procedures are almost the same.