The Family of Global Attractors of Coupled Kirchhoff Equations

In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior estimates of the equation in E0 and Ek space, and then the existence and uniqueness of solution is verified by Galerkin’s method. Then, the solution semigroup S(t) is defined, and the bounded absorptive set Bk is obtained on the basis of prior estimation. Through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek. Finally, by linearizing the equation, it is proved that the solution semigroup S(t) is Frechet differentiable on Ek, and the family of global attractors Ak have finite Hausdroff dimension and Fractal dimension.

The Existence of Global Attractor for Zakharov Equations Arising from Ion-acoustic Modes

In this paper,we consider the initial-boundary problem for Zakharov equations arising from ion-acoustic modes and obtain the existence of global attractor for the initialboundary problem for Zakharov equations.

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

The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.

A Family of the Global Attractor for Higher Order Nonlinear Kirchhoff Equation

In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations with strong damping terms. We will properly assume the stress term M(s) and nonlinear term g(ut). First, we can prove the existence and uniqueness of the solution of the equation via a prior estimate and Galerkin’s method, then the existence of the family of global attractor is obtained. At last, we can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractor are finite.

LONG TIME BEHAVIOR FOR SOLUTION OF INITIAL-BOUNDARY VALUE PROBLEM OF ONE CLASS OF SYSTEMS WITH MULTIDIMENSIONAL INHOMOGENEOUS GBBM EQUATIONS

The following initial-boundary value problem for the systems with multidimensional inhomogeneous generalized Benjamin-Bona-Mahony ( GBBM ) equations is reviewed. The existence of global attractors of this problem was proved by means of a uniform priori estimate for time.

The Global Attractor of Thermoelastic Coupled System

In this paper, we consider a class of Sine-Gordon equations which arise from the model of the thermoelastic coupled rod. Firstly, by virtue of the classical semigroup theory, we prove the existence and uniqueness of the mild solution under certain initial-boundary value for above-mentioned equations. Secondly, we obtain the boundedness of solutions by the priori estimates. Lastly, we prove the existence of a global attractor.

The Global Attractors for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Linear Damping and Source Terms

In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a nonlinear viscoelastic wave equation with strong damping, linear damping and source terms. Then we study the global attractors of the equation.

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.

Existence of Global Attractor for LS Type Equations

The global attractor problem of the long-short wave equations with periodic boundary condition was studied. The existence of global attractor of this problem was proved by means of a uniform a priori estimate for time.

Existence of Global Attractor for a 3D Brinkman-Forchheimer Equation in Some Poincaré Unbounded Domains

In this paper, we study the existence of global attractor of a class of three-dimensional Brinkman-Forchheimer equation in some unbounded domains which satisfies Poincaré inequality. We use the tail estimation method to establish the asymptotic compactness of the solution operator and then prove the existence of the global attractor in(H_0~1(Ω))~3.

Long Time Behavior of a Class of Generalized Beam-Kirchhoff Equations

In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making some assumptions about the nonlinear function term . The existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation are proved.

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 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 and Dimension Estimation for a 2D Generalized Anisotropy Kuramoto-Sivashinsky Equation

In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a two dimensional generalized anisotropy Kuramoto-Sivashinsky Equation. Then we prove the existence of the global attractor. Finally, we get the upper bound estimation of the Haus-dorff and fractal dimension of 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.

STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

In this article, we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov–Poisson–Fokker–Planck equations in the whole space by uniform-in-time energy estimates. The non-linear coupling of macroscopic part and Fokker–Planck operator in the model brings new difficulties for the energy estimates, which is resolved by adding tailored weighted-in-v energy estimates suitable for the Fokker–Planck operator.

Global Attractor of Two-Dimensional Strong Damping KDV Equation and Its Dimension Estimation

Firstly, a priori estimates are obtained for the existence and uniqueness of solutions of two dimensional KDV equations, and prove the existence of the global attractor, finally get the upper bound estimation of the Hausdorff and fractal dimension of attractors.

无界channel-like域上带有强阻尼项的Navier-Stokes方程的全局吸引子

本文主要研究定义在无界channel-like域上带有强阻尼项的Navier-Stokes方程解的渐近行为,为克服Sobolev嵌入在无界域中的非紧性,利用一致尾部估计方法在相空间中建立解的渐近紧性,并获得所研究问题全局吸引子的存在性。

一类广义BBM-Burgers方程的Cauchy问题

证明了广义BBM-Burgers方程的Cauchy问题vt-αvxxt-βvxx+γvxxxx+f(v)x=G(v)+h(vx)x+g(v)xx,x∈R,t>0,v(x,0)=v0(x),x∈R存在唯一整体强解v∈C([0,∞);Hs(R))∩C1([0,∞);Hs-2(R))(s≥4)和唯一的整体古典解,并给出解的衰减估计.

The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping

In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space E0 and Ek, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set B0k on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup S(t) of the equation has the family of the global attractor Ak in space Ek. Finally, we prove that the solution semigroup S(t) is Frechet differentiable on Ek via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor Ak.