Relation between Different Types of Global Attractors of Set-Valued Nonautonomous Dynamical Systems

The article is devoted to the study of the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems (cocycles). Here it is proved that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but we prove that every global pullback attractor of an α-condensing set-valued cocycle is always a local forward attractor. The obtained general results are applied while studying periodic and homogeneous systems. We give also a new criterion of the absolute asymptotic stability of nonstationary discrete linear inclusions. Dedicated to our friend Professor Enrico Primo Tomasini on the occasion of his 55th birthdayMathematics Subject Classifications (2000) Primary: 34C35, 34D20, 34D40, 34D45, 58F10,58F12, 58F39; secondary: 35B35, 35B40.     

On Attractors of Multivalued Semi-Flows and Differential Inclusions

In this paper we study the existence of global attractors for multivalued dynamical systems. These theorems are then applied to dynamical systems generated by differential inclusions for which the solution is not unique for a given initial state. Finally, some boundary-value problems are considered.     

带有热记忆的非均匀柔性结构的长时间动力行为

本文研究了带有热效应的非均匀柔性结构方程,并且该热效应符合Coleman-Gurtin定律.利用半群方法,建立了系统的整体适定性.主要结论是该系统的长时间动力行为.本文证明了系统的拟稳定性,整体吸引子的存在性以及整体吸引子具有有限的分形维数.此外,还证明了指数吸引子的存在性.     

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.     

具结构阻尼的拟线性膜方程的吸引子及其上半连续性

本文研究具结构阻尼的拟线性膜方程utt+△2u+(-△)αut+△φ(△u)+f(u)=g的适定性以及解的长时间动力学行为,其中α∈(1,2),旨在研究耗散指标α对方程解的适定性和长时间动力学行为的影响.本文证明非线性项φ(s)存在一个依赖于耗散指标α的临界指数pα=N+4(α-1)/(N-4(α-1))(N=3,4)...     

Global Attractors for Multivalued Random Dynamical Systems Generated by Random Differential Inclusions with Multiplicative Noise

In this paper we consider a stochastic differential inclusion with multiplicative noise. It is shown that it generates a multivalued random dynamical system for which there also exists a global random attractor.     

复Swift-Hohenberg方程在一些Banach空间内的指数吸引子

本文研究了N-维(N≤3)复Swift-Hohenberg方程在一些Banach空间x~α中解的渐近行为.运用Cholewa等人的技巧,证明了整体解的存在性以及整体吸引子A的存在性.最后,作为本文的另—个主要结果,证明了指数吸引子M的存在性,从而得到A有有限的分形维数.由于应用于Hilbert空间中所谓的挤压性质在我们的框架下不能成立,为了构造M,没有应用Hilbert空间中的标准的方法,而是应用Efendiev,Miranville,和Zelik最近的结果.     

Global exponential stability of almost periodic solution for a large class of delayed dynamical systems

Research on delayed neural networks with variable self-inhibitions, interconnection weights, and inputs is an important issue. In this paper, we discuss a large class of delayed dynamical systems with almost periodic self-inhibitions, inter-connection weights, and inputs. This model is universal and includes delayed systems with time-varying delays, distributed delays as well as combination of both. We prove that under some mild conditions, the system has a unique almost periodic solution, which is globally exponentially stable. We propose a new approach, which is independent of existing theory concerning with existence of almost periodic solution for dynamical systems.     

Attractors for General Operators

In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua.     

Trajectory attractors for nonclassical diffusion equations with fading memory

In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7,8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.     

Random attractor for stochastic lattice dynamical systems with α-stable Lévy noises

The present paper is devoted to the existence of a random attractor for stochastic lattice dynamical systems with α-stable Lévy noises.     

Dynamics of a conserved phase-field system

Recently, in Bonfoh [Ann. Mat. Pura Appl. 2011;190:105–144], we investigated the dynamics of a nonconserved phase-field system whose singular limit is the viscous Cahn–Hilliard equation. More precisely, we proved the existence of the global attractor, exponential attractors, and inertial manifolds and we showed their continuity with respect to a singular perturbation parameter. In the present paper, we extend most of these results to a conserved phase-field system whose singular limit is the nonviscous Cahn–Hilliard equation. These equations describe phase transition processes. Here, we give a direct proof of the existence of inertial manifolds that differs from our previous method that was based on introducing a change of variables and an auxiliary problem.