Global Dynamics of a Nonlinear Beam Equation with Strong Structural Damping

In this paper, the global attractor, exponential attractor and flat inertial manifold are obtained for a nonlinear beam equation with strong structural damping.     

Global random attractors for the stochastic dissipative Zakharov equations

The stochastic dissipative Zakharov equations with white noise are mainly investigated. The global random attractors endowed with usual topology for the stochastic dissipative Zakharov equations are obtained in the sense of usual norm. The method is to transform the stochastic equations into the corresponding partial differential equations with random coefficients by Ornstein-Uhlenbeck process. The crucial compactness of the global random attractors wiil be obtained by decomposition of solutions.     

Three counterexamples in the theory of inertial manifolds

An example of a dissipative semilinear parabolic equation in a Hilbert space without smooth inertial manifolds is constructed. Moreover, the attractor of this equation can be embedded in no finite-dimensionalC ¹ invariant submanifold of the phase space. The class of scalar reaction-diffusion equations in bounded domains Ω ⊂ ℝ^m without inertial manifolds with the property of absolute normal hyperbolicity on the setE of stationary points of the phase semiflow is described. Such equations may have inertial manifolds with the weaker property of normal hyperbolicity onE. Three-dimensional reaction-diffusion systems without inertial manifolds normally hyperbolic at stationary points are found. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 439–447, September, 2000.     

一类非线性抛物型变分不等方程的全局吸引子及弱近似惯性流形

对由一类非线性抛物型变分不等方程所描述的无穷维动力系统，给出了存在全局吸引子及弱近似惯性流形的充分条件．     

Cahn－Hiliard方程的近似惯性流形

通过压缩映象原理，本文对Ｃａｈｎ－Ｈｉｌｉａｒｄ方程构造了一簇近似惯性流形，这些近似惯性流形越来越逼近全局吸引子     

非自治Ginzburg-Landau方程的周期解和全局周期吸引子

研究受周期外力影响的非自治Ginzburg-Landau方程的解的长时间行为.首先证明系统在空间H上存在周期解,而且周期解包含在空间V中的一个有界吸收集内.然后证明了当耗散系数λ满足一定条件时,该系统在空间H上具有唯一的周期解,该周期解指数吸引H中的任意有界集.     

On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions

In this paper we define multivalued semiprocesses and give theorems providing the existence of global attractors for such systems. This theory generalizes the construction of nonautonomous dynamical systems given by V. V. Chepyzhov and M. I. Vishik to the case where the system is not supposed to have a unique solution for each initial state. Further, we apply these theorems to nonautonomous differential inclusions of reaction–diffusion type.     

弱D-拉回指数吸引子的存在性

主要研究弱D-拉回指数吸引子的存在性.首先讨论了弱D-拉回指数吸引子与非紧性测度之间的关系,其次,建立了弱D-拉回指数吸引子存在性的一般方法,最后证明了外力项具有指数增长速度的反应扩散方程在H_0~1(Ω)中存在弱D-拉回指数吸引子.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

Global Periodic Attractor for Strongly Damped and Driven Wave Equations

In this paper we consider the strongly damped and driven nonlinear wave equations under homogeneous Dirichlet boundary conditions. By introducing a new norm which is equivalent to the usual norm, we obtain the existence of a global periodic attractor attracting any bounded set exponentially in the phase space, which implies that the system behaves exactly as a one dimensional system.     

Global Attractors for a Nonclassical Diffusion Equation

We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of arbitrary order.     

强阻尼波动方程吸引子的正则性及其逼近

该文研究强阻尼波动方程的初边值问题．利用线性主算子在相空间中生成的解析半群的性质,证明了解的光滑效应,这个现象与弱阻尼波动方程的情形大不相同．由此作者得到了吸引子的正则性,并象自伴情形那样构造了近似惯性流形．     

带衰退记忆的经典反应扩散方程的强全局吸引子*

当任意阶多项式增长的非线性项为耗散,且外力项仅属于L~2(Ω)时,研究了带衰退记忆的经典反应扩散方程的解在强拓扑空间H_0~1(Ω)×L_μ~2(R~+;D(A))的长时间行为.应用抽象函数理论、半群理论以及新的估计技巧,在拓扑空间H_0~1(Ω)×L_μ~2(R~+;D(A))上,验证了强解半群的渐近紧性并且证明了强全局吸引子的存在性.     

非齐次边界条件下Sine-Gordon型二阶非线性系统的全局吸引子及能稳性

本文在非齐次边界条件下,证明了有阻尼Sine-Gordon型二阶非线性系统的全局吸引子的存在性．同时证明了当时,该全局吸引子为—个平衡点,而且该平衡点具有指数吸引性．并将所得结果应用到系统能稳性的研究中．     

一类具有Leakage时滞的惯性Cohen-Grossberg神经网络的全局指数稳定性和Hopf分支

研究一类具有Leakage时滞的惯性Cohen-Grossberg神经网络模型.通过构造适当的Lyapunov泛函得到了平衡点全局指数稳定的充分条件.通过分析特征方程,讨论了系统平衡点的局部稳定性,得出了系统Hopf分支存在的充分条件.最后对所得理论结果进行了数值模拟.     

Pullback Attractors of Nonautonomous and Stochastic Multivalued Dynamical Systems

In this paper we study the existence of pullback global attractors for multivalued processes generated by differential inclusions. First, we define multivalued dynamical processes, prove abstract results on the existence of -limit sets and global attractors, and study their topological properties (compactness, connectedness). Further, we apply the abstract results to nonautonomous differential inclusions of the reaction–diffusion type in which the forcing term can grow polynomially in time, and to stochastic differential inclusions as well.