A Family of Global Attractors for the Generalized Kirchhoff-Beam Equations

In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality.

Global Attractor of a Reaction-diffusion System with Hamiltonian Structure

In this paper, we consider a reaction diffusion system with Hamitonian structure, we first prove the existence of an invariant region for system and the continuity of the semigroup, then establish the absorbing sets and global attractor.

Global attractor of 2D autonomous g-Navier-Stokes equations

In this paper, the existence of global attractors for the 2D autonomous g- Navier-Stokes equations on multi-connected bounded domains is investigated under the general assumptions of boundaries. The estimation of the Hausdorff dimensions for global attractors is given.

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.

ATTRACTORS FOR FULLY DISCRETE FINITE DIFFERENCE SCHEME OF DISSIPATIVE ZAKHAROV EQUATIONS

A fully discrete finite difference scheme for dissipative Zakharov equations is analyzed. On the basis of a series of the time-uniform priori estimates of the difference solutions, the stability of the difference scheme and the error bounds of optimal order of the difference solutions are obtained in L^2 × H^1 × H^2 over a finite time interval （0, T]. Finally, the existence of a global attractor is proved for a discrete dynamical system associated with the fully discrete finite difference scheme.

GLOBAL ATTRACTOR FOR CAMASSA-HOLM TYPE EQUATIONS WITH DISSIPATIVE TERM

In this paper, the global existence of classical solution and global attractor for Camassa-Holm type equations with dissipative term are established by using fixed point theorem and a priori estimates.

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE H^k

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.

GLOBAL ATTRACTOR FOR HASEGAWA-MIMA EQUATION

The long time behavior of solution of the Hasegawa-Mima equation with dissipation term was considered. The global attractor problem of the Hasegawa-Mima equation with initial periodic boundary condition was studied. Applying the uniform a priori estimates method, the existence of global attractor of this problem was proved, and also the dimensions of the global attractor was estimated.

Global Attractors for Nonclassical Diffusion Equations of Kirchhoff Type

In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions are proved. At last, the existence of global attractors and its upper semicontinuous property are discussed.

THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton （gμHS2） system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative （gμHS2） system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative （gμHS2） system, we show that the semi-group of the solution operator {S（t）}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S（t）}t≥0 possesses a global attractor in the Sobolev space H2（S） × H2（S）.

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hper^k is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hper^k.

GLOBAL ATTRACTOR OF A CLASS OF STRONGLY DAMPED NONLINEAR WAVE EQUATIONS

A class of strongly damped nonlinear wave equations are studied by using the technique of the operator decomposition,and the existence of the global compact attractor in space D(A)×V is obtained.

A Family of Global Attractors for a Class of Generalized Kirchhoff-Beam Equations

The initial boundary value problem for a class of high-order Beam equations with quasilinear and strongly damped terms is studied. Firstly, the existence and uniqueness of the global solution of the equation are proved by prior estimation and Galerkin finite element method. Then the bounded absorption set is obtained by prior estimation, and the family of global attractors for the high-order Kirchhoff-Beam equation is obtained. The Frechet differentiability of the solution semigroup is proved after the linearization of the equation, and the decay of the volume element of the linearization problem is further proved. Finally, the Hausdorff dimension and Fractal dimension of the family of global attractors are proved to be finite.

The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.

Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model

In this paper, we study the long time behavior of solution to the initial boundary value problem for a class of Kirchhoff-Boussinesq model flow . We first prove the wellness of the solutions. Then we establish the existence of global attractor.

Global Attractors for a Class of Generalized Nonlinear Kirchhoff-Sine-Gordon Equation

In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.

Regularity and finite dimensionality of attractor for plate equation on R^n

This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlier work of the author. It is shown that the attractor is actually a bounded set of the phase space and has finite fractal dimensionality.

A REMARK ON THE ATTRACTORS OF NON-NEWTONIAN FLUIDS

This paper corrects some mistakes in the proof of absorbing sets theorem of at-tractors of non-Newtonian fluids, and establishes again the existence of bounded absorbing sets.