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 Attractor and Inertial Manifolds for Regeneration of Severed Limb Equation

In this paper the existence of the compact global attractor and inertial manifolds for regeneration of severed limb equation are proved.     

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.     

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.     

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 the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

We show the existence of an inertial manifold (ie, a globally invariant, exponentially attracting, finite‐dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.     

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.     

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.     

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.     

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.     

The existence of random $\mathcal{D}$-pullback attractors for random dynamical system and its applications

In this paper, we establish a result on the existence of random $\mathcal{D}$-pullback attractors for norm-to-weak continuous non-autonomous random dynamical system. Then we give a method to prove the existence of random $\mathcal{D}$-pullback attractors. As an application, we prove that the non-autonomous stochastic reaction diffusion equation possesses a random $\mathcal{D}$-pullback attractor in $H_0^1$ with polynomial growth of the nonlinear term.     

Global Well-posedness and Global Attractor for Two-dimensional Zakharov-Kuznetsov Equation

The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in H^s(R²) for all 5/7 < s < 1 via using I-method in the context of atomic spaces. By means of the increment of modified energy, the existence of global attractor for the weakly damped, forced Zakharov-Kuznetsov equation is also established in H^s(R²) for 10/11 < s < 1.     

A Description of the Global Attractor for a Class of Reaction – Diffusion Systems with Periodic Solutions

We examine the autonomous reaction–diffusion system with Dirichlet boundary conditions on (0, 1), where α, β are real, α > 0, and g is C¹ and satisfies some conditions which we need in order to prove the existence of solutions. We construct a solution of (RD) for every initial value in L²((0, 1)) × L²((0, 1)), we show that this solution is uniquely determined and that the solution has C^∞–smooth representatives for all positive t. We determine the long time behaviour of each solution. In particular, we show that each solution of (RD) tends either to the zero solution or to a periodic orbit. We construct all periodic orbits and show that their number is always finite. It turns out that the global attractor is a finite union of subsets of L² × L², which are finite–dimensional manifolds, and the dynamics in these sets can be described completely.     

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.