A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.

The Family of Exponential Attractors and Inertial Manifolds for a Generalized Nonlinear Kirchhoff Equations

In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space E0 to Ek, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.

A Family of Exponential Attractors and Inertial Manifolds for a Class of Higher Order Kirchhoff Equations

In this paper, the global dynamics of a class of higher order nonlinear Kirchhoff equations under n-dimensional conditions is studied. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup associated with the initial boundary value problem are proved, and the existence of a family of exponential attractors is obtained. Then, by constructing the corresponding graph norm, the condition of a spectral interval is established when N is sufficiently large. Finally, the existence of the family of inertial manifolds is obtained.

The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations

To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets Bk. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system;thereby we obtain a family of random attractors.

PULLBACK EXPONENTIAL ATTRACTORS FOR THE NON-AUTONOMOUS MICROPOLAR FLUID FLOWS

This paper investigates the pullback asymptotic behaviors for the non-autonomous micropolar fluid flows in 2D bounded domains. We use the energy method, combining with some important properties of the generated processes, to prove the existence of pullback exponential attractors and global pullback attractors and show that they both with finite fractal dimension. Further, we give the relationship between global pullback attractors and pullback exponential attractors.

EXPONENTIAL ATTRACTORS FOR A GENERALIZED GINZBURG-LANDAU EQUATION

Based on the paper [1]. we obtain the existence of exponential attractors for a generalized Ginzburg-Landau equation in one dimension

Exponential Attractors for the Ginzburg-Landau-BBM Equations

In this paper, the existence of the exponential attractors for the Ginzburg- Landau-BBM equations with periodic initial and boundary conditions are obtained by using the squeezing property and the operator dccomposition method.

Exponential Attractors for the Generalized Ginzburg-Landau Equation

In this paper,we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions.We show the squeezing property and the existence of fimte dimen- sional exponential attractors for this equation

Uniform Exponential Attractors for Second Order Non-autonomous Lattice Dynamical Systems

In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, a-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.

EXPONENTIAL ATTRACTORS OF THE GINZBURG-LANDAU-BBM EQUATIONS IN AN UNBOUNDED DOMAIN

In this paper, the existence of the exponential attractors for the Ginzburg-Landau-BBM equations in an unbounded domain is proved by using weighted function and squeezing property.

EXPONENTIAL ATTRACTOR FOR THE GENERALIZED SYMMETRIC REGULARIZED LONG WAVE EQUATION WITH DAMPING TERM

The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.

Long-Time Behavior of Solution for Autonomous Suspension Bridge Equations with State-Dependent Delay

This work is devoted to the following suspension bridge with state-dependent delay: . The main goal of this paper is to investigate the long-time behavior of the system. Under suitable hypothesis, the quasi-stability estimates of the system are established, based on which the existence of global attractor with finite fractal dimension is obtained. Furthermore, the existence of exponential attractor is proved.

LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW

This study addresses long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law,the convolution kernel of which entails an extremely weak dissipation in the thermal law.By using the semigroup theory,we first establish the existence of global weak and strong solutions as well as their continuous dependence on the initial data in appropriate function spaces,under suitable assumptions on the weight of time delay term,the external force term and the nonlinear term.We then prove that the system is quasi-stable and has a gradient on bounded variant sets,and obtain the existence of a global attractor whose fractal dimension is finite.A result on the exponential attractor of the system is also proved.

Exponential Attractor of the 3D Derivative Ginzburg-Landau Equation

In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved.

Exponential Attractor for Davey-Stewartson Equation in a Banach Space

In this paper, the existence of the exponential attractor of Davey-Stewartson equation is considered and its estimation of fractal dimension is obtained in a Banach subspace Xp^α of L^p（Ω）.

Uniform Exponential Attractor for Nonautonomous Partly Dissipative Lattice Dynamical System

In this paper,we consider the existence of a uniform exponential attractor for the nonautonomous partly dissipative lattice dynamical system with quasiperiodic external forces.

Exponential Attractor for a Nonlinear Boussinesq Equation

This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space H0^2（0, 1） × L^2（0, 1）. The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space H0^3（0, 1） × H0^1（0, 1）.

The Weighted Estimates of the Schrdinger Operators on the Nilpotent Lie Group

In this paper,we consider the strong dissipative KDV type equation on an unbounded domain R1.By applying the theory of decomposing operator and the method of constructing some compact operator in weighted space,the existence of exponential attractor in phase space H2（R1） is obtained.