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

estimation for the higher-order nonlinear Kirchhoff-type equation* Ling Chen, Wei Wang, Guoguang Lin Ling Chen,Department of Mathematics, Yunnan University, Kunming, Yunnan 650091 chenl800828@163.com Wei Wang,Department of Mathematics, Yunnan University, Kunming, Yunnan 650091 wangw2641@163.com Guoguang Lin,Department of Mathematics, Yunnan University, Kunming, Yunnan 650091 gglin@ynu.edu.cn Abstract We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-type

. Under of the proper assume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkin method, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of the global attractor.

Introduction
We consider the problem is a strongly dissipation.
There have been many researches on the global attractors existence of the Kirchhoff equation with strong dissipation, we can see [1,2,3]. There are lots of recent results on the global attractor of Kirchhoff equation, we can refer [4,5,6,7].
Zhijian Yang and Pengyan Ding [8] studied the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on   Recently, Zhijian Yang, Pengyan Ding and Lei Li [9] also studied longtime dynamics of the Kirchhoff equation with fractional damping and supercritical nonlinearity:  is a bounded domain in n R with the smooth boundary, they show that (i) even if p (the growth exponent p of the nonlinearity () fu), , the limit solutions exist and possesses a weak global attractor.
Chueshov [10] first studied the well-posedness and the global attractor for the IBVP of Kirchhoff wave models with strong nonlinear damping: He established a finite-dimensional global attractor in the sense of partially strong topology. In particular, in nonsupercritical case: (i) the partially strong topology becomes strong; (ii) an exponential attractor is obtained in natural energy is also a constant.  (2) Then the solution ( , ) uv of the problems(1. (2.   ( ( ) ( ) , ) From the above ,we have Where we take proper constant 0 m and  , such that: Then we take (2.12) By using Gronwall inequality, we obtain: where I S S N 2 3 4 7 -1 9 2 1 V o l u m e 1 2 N u m b e r 0 9 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s (2.14) So, we have: Thus there exist 1 () tt  and 0 R , such that (2.17) Then the solution ( , ) uv of the problems(1. multiply both sides of equation (1.1) and obtain (2.20) According to assume (1), we can get Next, accord to assume (2), we see By using Gronwall inequality, we obtain , the procedure is omitted . Next, we prove the uniqueness of solution in detail.
Let , uv are two solutions of equation (1.1), we denote w u v , then two equations subtract and obtain By using t w to inner product of the equation (2.33), and we have According to (2.37) -(2.39), we have

Acknowledgements
We express our sincere thanks to the anonymous reviewer for his/her careful reading of the paper, we hope that we can get valuable comments and suggestions. Making the paper better.