The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation with nonlinear strongly damped terms*

In this paper ,we study the long time behavior of solution to the initial boundary value problems for higher -order kirchhoff-type equation with nonlinear strongly dissipation: u ( ) ( ) ( ) ( ) ( ). 2 u u u h u f x t m q m t m tt            At first ,we prove the existence and uniqueness of the solution by priori estimate and Galerkin methodthen we establish the existence of global attractors ,at last,we consider that estimation of upper bounds of Hausdorff and fractal dimensions for the global attractors are obtain.


Introduction
In this paper we concerned with the long time behavior of solution to the initial boundary value problems for Higher-order Kirchhoff-type equation with nonlinear strongly dissipation : There have been many researches on the well-positive and the longtime dynamics for Kirchhoff equation.we can see [1][2][3][4][5][6],FUCAI Li [5] deals with the higher-order kirchhoff-type equation with nonlinear dissipation: (1. 6 ,then for any initial data with negative initial energy,the solution blows up at finite time in 2 L  p norm. Yang Zhijian, Wang Yunqing [6] also studied the global attractor for the Kirchhoff type equation with a strong dissipation: (1.9) is an extrnal force term.it proves that the relative continuous semigroup ) ( t S possesses in the phase space with low regularity a global attractor which is connected.
Yang zhijian, Cheng Jianling [7] studies the asymptotic behavior of solutions to the Kirchhoff-type equation: They prove that the related continuous semigroup

Preliminaries
For convenience,we denote the norm and scalar product in ) ( 2  L by . and (.,.); In this section, we present some materials needed in the proof of our results, state a global existence result, and prove our main result. For this reason, we assume that and notations needed in the proof of our results.For this reason, we assume that there exist ,then the solution we multiply v with both sides of equation (1.1) and obtain ) ), For above ,we have By using Poincare inequality,we obtain: By using Young's inequality,we obtain ( 2.14) ,and 1   q p using Young's inequality.we 0btain From (2.12), we know (2.28) (2.29) So,there exist

the existence and uniqueness of solution
Then two equations subtract and obtain According to Lemma1, Lemma2,we have According to Young's inequality, we get Form above ,we have (3.12) According to

Global attractor
2) It exists a bounded absorbing set 3) When 0 , S (t) t  is a completely continuous operator A .
Therefore , the semigroup operator () St exists a compact global attractor.  (2) Furthermore, for any

The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor
We rewrite the problems (1.1.)-(1.3): is a continuous mapping,satisfy the follow conditions.
s Frechet differentiable ,it exists is a bounded linear differential operator  ) , ( .that is The proof of lemma 4.1 see ref. [9],is omitted here .According to Lemma ,is an isomorphic mapping.So let is the global attractor of is also the global attractor of (t)} {S  ,then  satisfies as follows: .
The initial condition (4.3) can be written in the following form: We take N n  ,then consider the corresponding n solution: , here u is the solution of problems (4.1);  represents the outer product , Then there is a