Global Attractors for the One-dimensional Model of Thermodiffusion with Second Sound

In this paper we investigate the global attractors for the one-dimensional linear model of thermodiffusion with second sound. Using the method of contractive functions, we obtain the asymptotically compact of the semigroup and the existence of the global

Notes on the Global Attractors for Semigroup

First we introduce two necessary and sufficient conditions which ensure the existence of the global attractors for semigroup. Then we recall the concept of measure of noncompactness of a set and recapitulate its basic properties. Finally, we prove that these two conditions are equivalent directly.

Global attractors associated with general partly dissipative reaction-diffusion systems in unbounded domains

The asymptotic behaviour of solutions for general partly dissipative reaction-diffusion systems in Rn is studied. The asymptotic compactness of the solutions and then the existence of the global attractor are proved in L2(Rn )× L2(Rn ) .

Global Attractors for the Initial Value Problems of Cohn-Hilliard Equations on Compact Manifolds

In this paper we establish some theorems which are concerned with the equivalent norms of Sobolev spaces on a Riemannian manifold. Using the theorems we prove the existence of global attractors for the initial value problem of Cahn-Hilliard equations. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors are also obtained.

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.

Global Existence, Asymptotic Behavior and Uniform Attractors for Damped Timoshenko Systems

In this paper, we consider the Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. And then we establish the global existence and asymptotic behavior of solution by using the semigroup method and multiplicative techniques, then further prove the existence of a uniform attractor by using the method of uniform contractive function. The main advantage of this method is that we need only to verify compactness condition with the same type of energy estimate as that for establishing absorbing sets.

Global Existence, Asympotic Behavior and Uniform Attractors for Thermoelastic Systems

In this paper, we first establish the global existence and asymptotic behavior of solutions by multiplicative techniques, then further prove the existence of a uniform attractor for a thermoelastic system by using the method of uniform contractive functions. We finally investigate an alternative result of solutions for the semilinear thermoelastic systems by virtue of the semigroup method.

奇异扰动的分数阶随机反应扩散方程的吸引子

研究了定义在光滑有界域O■R^(n)上、带乘性噪声和奇异扰动的分数阶随机反应扩散方程在L^(2)(O)上的随机拉回吸引子的存在性。由于奇异扰动和分数阶算子对系统的作用,无法用正则估计得到系统的紧性问题,因此利用Aubin紧性定理获得了由方程生成的随机动力系统的渐近紧性。

Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains

In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.

Random Attractors of Stochastic Non-Autonomous Nonclassical Diffusion Equations with Linear Memory on a Bounded Domain

In this article, we discuss the long-time dynamical behavior of the stochastic non-autonomous nonclassical diffusion equations with linear memory and additive white noise in the weak topological space . By decomposition method of the solution, we give the necessary condition of asymptotic compactness of the solutions, and then prove the existence of random attractor, while the time-dependent forcing term only satisfies an integral condition.

Attractors for the Stochastic Lattice Selkov Equations with Additive Noises

In this paper, by proving the pullback asymptotic compactness of the stochastic lattice Selkov equations with the cubic nonlinearity, the existence of a random attractor of the stochastic lattice reversible Selkov equations on infinite lattice with additive noises is proved.

More Compactification for Differential Systems

This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.

Existence and Upper Semi-Continuity of Random Attractors for Nonclassical Diffusion Equation with Multiplicative Noise on Rn

This paper is concerned with the existence and upper semi-continuity of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity and multiplicative noise in H1(Rn). First, we study the existence and uniqueness of solutions by a noise arising in a continuous random dynamical system and the asymptotic compactness is established by using uniform tail estimate technique, and then the existence of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity. As a motivation of our results, we prove an existence and upper semi-continuity of random attractors with respect to the nonlinearity that enters the system together with the noise.

含非线性阻尼的2D g-Navier-Stokes系统解的全局吸引子及渐近光滑效应

Navier-Stokes方程作为流体动力学研究的基本方程之一,描述了作用于液体任意给定区域的力的动态平衡,反映了粘性流体流动的基本力学规律,在流体力学中具有十分重要的意义。作为Navier-Stokes方程的一种推广,近年来有关g-Navier-Stokes方程的研究工作方兴未艾。针对一类含非线性阻尼的自治g-Navier-Stokes系统的动力学性质展开研究,借助解的先验估计和能量方程方法,在R2上证明了解的全局渐近紧性,得到解半群全局吸引子的存在性,并对解的渐近光滑效应进行了分析,进一步推广且改进了近年来已有的部分经典研究成果,丰富了g-Navier-Stokes方程的相关研究理论。

Two-Dimensional Riemann Problems:Transonic Shock Waves and Free Boundary Problems

We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

带有分布导数的反应扩散方程在Rn中全局吸引子的存在性

该文考虑R^(n)中一类带有加权项V(x)和分布导数的反应扩散方程解的长时间行为,证明了(L^(2)(R^(n)),L^(2)(R^(n))∩Lp(R^(n)))-全局吸引子的存在性;而且对任意的δ1∈[0,+∞),该吸引子能够在L^(2)∩Lp+δ1-拓扑下吸引L^(2)(R^(n))中的有界集.

一类六阶非线性扩散方程的整体吸引子

研究一类由具有二阶导数项的Landau-Ginzburg自由能量泛函导出的六阶非线性扩散方程初边值问题的整体动力学行为.基于解的一致估计以及算子半群的渐近紧性证明了当初值u_(0)∈H^(2)_(per)(Ω)时,方程所张成的算子半群在H^(6)_(per)(Ω)存在一个整体吸引子.该研究结果可以使我们更好地了解六阶非线性扩散方程解长时间行为,为研究两相物质之间相互扩散现象提供理论基础.

Plate方程时间依赖全局吸引子的存在性

时间依赖全局吸引子的概念是Di Plinio,Temam等人于近期提出的.在非线性项f满足临界增长条件时,应用算子分解技巧,验证了系数参数与时间t有关时plate方程对应的过程族{U(t,τ)}的渐近紧性,进而获得了plate方程时间依赖全局吸引子的存在性及正则性.

一类非线性发展方程解的长时间行为

研究一类非线性发展方程整体弱解的长时间行为,首先利用算子分解方法证明了系统整体弱解的渐近正则性,由此得到了整体弱解对应的解半群在H1(R3)×H1(R3)的全局吸引子A的存在性,然后证得A在H2(R3)×H2(R3)有界,其中非线性项满足临界指数增长.

一类非线性发展方程的渐近吸引子

无穷维动力系统的基本理念是将一个无穷维系统约化为一个有限维系统,但是,要进一步研究约化后的有限维系统的动力学行为是非常困难的,因为它们的结构是未知的。为了克服这个困难,诸如近似惯性流形等概念已被引入,对于Navier-Stokes方程,其近似惯性流形的存在性问题已被讨论,它是通过挤压性质找到一个Lipschitz函数,说明其整体吸引子位于该函数图的某个小领域,而文中是通过构造一个有限维解序列,说明长时间后其趋于方程的整体吸引子,理论上给出了一类发展方程的渐近吸引子的构造方法.