ASYMPTOTIC BEHAVIOR OF WAVE EQUATION OF KIRCHHOFF TYPE WITH STRONG DAMPING

The paper deals with the strongly damped nonlinear wave equation of Kirchhoff type. The existence of a global attractor is proven by using the decomposition, and moreover, the structure of the global attractor is established. Our results improve the previous results.     

Long-Time Dynamics for a Nonlinear Viscoelastic Kirchhoff Plate Equation

This paper is devoted to study the long-time dynamics for a nonlinear viscoelastic Kirchhoff plate equation. Under some growth conditions of g and f, the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.     

单模光纤中光孤立子传播的长距离行为(英文)

研究描述单模光纤中光孤立子传播的具光纤损耗项的三阶非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程,首先证明了整体解的存在唯一性结果,然后证明其长距离行为由紧的整体吸引子刻画,并给出了吸引子的Ｈａｕｓｄｏｒｆｆ维数和分形维数的上界估计,最后研究了吸引子的正则性．     

非线性Sobolev-Galpern方程的有限维整体吸引子

本文研究非线性Sobolev-Galpern方程解的渐近性态．首先证明了该方程在H^2(Ω)∩H0^1(Ω)中整体弱吸引子的存在性，然后利用一个能量方程证明了整体弱吸引子实际上是整体强吸引子，建立了整体吸引子的有限维性．     

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.     

一类具有非线性kirchhoff-sine-Gordon广义方程的整体吸引子的存在性

主要目的是利用Galerkin逼近法和先验估计来证明一类具有非线性阻尼和外源项的耗散型sine-Gordon-kirchhoff方程的整体吸引子的存在性,首先通过先验估计证明系统存在唯一的整体解,再证明系统存在有界吸收集和算子半群光滑性质,最后得到系统存在整体吸引子.     

Large Time Behavior via the Method of ?-Trajectories

The method of ?-trajectories is presented in a general setting as an alternative approach to the study of the large-time behavior of nonlinear evolutionary systems. It can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear. Here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics. In both cases we prove the existence of a finite-dimensional global attractor and the existence of an exponential attractor.     

Attractors and dimension of dissipative lattice systems

In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541-1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342-368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction-diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin.     

Global and exponential attractors for mixtures of solids with Fourier’s law

This paper presents a study of the long-time dynamics of the dynamical system generated by a nonlinear system modeling mixture of solids with nonlinear damping and Fourier’s law. By using the recent quasi-stability theory, we prove the existence of a smooth finite dimensional global attractor, which is characterized as an unstable manifold of the set of stationary solutions. The quasi-stability of the system is achieved through an estimated stabilizability. Moreover, the existence of a generalized exponential attractor is shown.     

具有双记忆项的热弹耦合梁方程的整体吸引子

研究了具有双记忆项的非线性热弹耦合梁方程,利用已知的研究结果给出解的适定性定理,其次通过先验估计并结合常用不等式证明系统存在有界吸收集,且利用标准方法验证半群的渐近紧性,得到整体吸引子的存在性.     

一类广义耦合的非线性波动方程组在无界区域上的整体吸引子

该文研究了一类广义耦合的非线性波动方程组在无界区域上的 整体吸引子问题. 利用插值不等式和加权空间, 得到了一系列关于时间的一致先验估计，证 明了整体吸引子的存在性.     

Attractor and dimension for strongly damped nonlinear wave equation

1. IntroductionConsider the strongly damped nonlinear wave equationwith the Dirichlet boundary conditionand the initial value conditionswhere u = u(x, t) is a real--valued function on fi x [0, co), fi is an open bounded set of R"with smooth boundary off, or > 0, g e L'(fl), D(--Q) ~ Ha(~~) n H'(fl).We assume for the function f(u) as follows'f(u) E CI (R, R) satisfiesfor any ig al, uZ E R, where k, hi > 0, i ~ 0, 1, 2, 61 > 0 and 0 5 6o < 1'The type of equation (1) can be regarded as the…     

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF THREE-DIMENSIONAL NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED

The three-dimensional nonlinear SchrSdinger equation with weakly damped that possesses a global attractor are considered. The dynamical properties of the discrete dynamical system which generate by a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete dynamical system.     

Ginzburg－Landau－Newell模型的动力学行为

木文对Ｇｉｎｚｂｕｒｇ－Ｌａｎｄａｕ－Ｎｅｗｅｄ模型的动力学行为进行了讨论，得到了该模型的整体吸引子的存在性，同时得到了此吸引子维数的下界估计和该吸引子的Ｈａｕｓｄｏｒｆｆ维数和Ｗａｃｔａｌ（分形）维数的上界估计．     

Long-time dynamics for a nonlinear Timoshenko system with delay

This paper is concerned with a nonlinear Timoshenko system with a time delay term in the internal feedback together with initial data and Dirichet boundary conditions. Under some suitable assumptions on the weights of feedback, we obtain the existence of a global attractor with finite fractal dimension for the case of equal speed wave propagation. Furthermore, the existence of exponential attractors is also derived.     

Long time behavior for damped generalized coupled nonlinear wave equations. Ⅱ.Unbounded domain

Introduction In [if, the authors established the unique existence of the smooth solution for the couplednonlinear equations. These were proposed to describe the interaction process of internal longwaves. In [2], Ito proposed a recursion operator by which he inferred that the equations possessinfinitely many symmetries and constants of motion. In [31, He established the existence ofa smooth solution to the system of coupled nonlinear KdV equations[41. We remark that'Schonbeh[sl dealt with a …     

Global attractors for von Karman evolutions with a nonlinear boundary dissipation

Dynamic von Karman equations with a nonlinear boundary dissipation are considered. Questions related to long time behaviour, existence and structure of global attractors are studied. It is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global (compact) attractor for all weak (finite energy) solutions. This result has been known in the case of full interior dissipation, but it is new in the case when the boundary damping is the main dissipative mechanism in the system. In addition, we prove that fractal dimension of the attractor is finite. The proofs depend critically on the infinite speed of propagation associated with the von Karman model considered.     

The finite dimensional behaviour for the higher-order nonlinear Ginzburg-Landau system inn spatial dimensions

In this paper, we study the 2m-order nonlinear Ginzburg-Landau system inn spatial dimensions. We show the existence and uniqueness of the global generalized solution, and the existence of the global attractor for this system, and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor. This project is supported by the National Natural Science Foundation of China (No. 19571010).     

Long-time behavior for a nonlinear plate equation with thermal memory

We consider a nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we use a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term f is real analytic. Moreover, we provide an estimate on the convergence rate.     

Attractors for autonomous double‐diffusive convection systems based on Brinkman–Forchheimer equations

In this paper, we consider the existence of global attractor and exponential attractor for some dynamical system generated by nonlinear parabolic equations in bounded domains with the dimension N≤4 which describe double‐diffusive convection phenomena in a porous medium. We deal with both of homogeneous Dirichlet and Neumann boundary condition cases. Especially, when Neumann condition is imposed, we need some assumptions and restrictions for the external forces and the average of initial data, since the mass conservation law holds. Copyright © 2015 John Wiley & Sons, Ltd.