Estimate on the Dimension of Global Attractor for Nonlinear Higher-Order Coupled Kirchhoff Type Equations

In this paper, we investigate the finite dimensions of the global attractor for nonlinear higher-order coupled Kirchhoff type equations with strong linear damping in Hilbert spaces 0 E and 1 E . Under the appropriate assumptions, we acquire a precise estimate of the upper bound for its Hausdorff and Fractal dimensions.


Introduction
G. G. Lin and L. J., Hu have studied the existence of a global attractor for coupled Kirchhoff type equations with strongly linear damping in [1].In this paper, we are concerned with the finite dimensions of the global attractor as mentioned above: , in 0,+ , , 0 , , 0 , , , 0 , , 0 , , 0, 0, 0,1, 2, , 1, , 0, where Ω is a bounded domain in n R with smooth boundary ∂Ω , In demonstrating the longtime behavior of evolutional equation, we currently aim to show that the dynamics of the equation is finite dimensional.To be precise, one possible way to express it is to say that the dynamical systems of equation exists a global attractor with finite Hausdorff and Fractal dimensions.
Concerning the wave equation with linear and semi-linear dissipative system, existence of the global attractor with finite Hausdorff and Fractal dimensions is proved in [2], for the nonlinear wave equation, the existence of the global attractor with finite Hausdorff and Fractal dimensions is proved in [3] [4] [5].
When the equation is nonlinear, the process of dimension estimation is more complicated.The method of linearization works very well on it, and meanwhile we take fully consideration of assumptions on the nonlinearities of the equation.
Recently, Z. J, Yang [6] studied the longtime behavior of the Kirchhoff type equation with strong damping on N R .It showed that the related continuous semi-group ( )

S t possesses a global attractor which is connected and has finite
Fractal and Hausdorff dimensions. ( , in .
At the same time, Z. J. Yang [7] dealt with the global attractors and their Hausdorff dimensions for a class of Kirchhoff models, and got the existence, regularity, and Hausdorff dimensions of global attractors for a class of Kirchhoff models arising in elastoplastic flow.
Furthermore, X. M. Fan and S. F. Zhou [8] proved the existence of compact kernel sections for the process generated by strongly damped wave equations of non-degenerate Kirchhoff type modelling the nonlinear vibrations of an elastic string, and they obtained a precise estimate of upper bound of Hausdorff dimension of kernel sections.
( ) In addition, G. G. Lin and Y. L. Gao [9] studied the longtime behavior of solution to initial boundary value problem for a class of strongly damped higher-order Kirchhoff type equation:

Preliminaries and Main Results
Throughout this paper, we need some notations for convenience.We consider a family of Hilbert spaces ∈ , whose inner products and norms are given by ( ) ( ) , , For our purpose, we define a weighted inner product and norm in 0 E by , , , , , , , u p v q u p v q E ϕ ϕ = = ∈ .Next, we make the following assumptions for problem (1.1)- (1.5).
is not decreasing function and for positive con- , and for every 0 R , there exist ( )

The Hausdorff and Fractal Dimensions of the Attractor
In order to obtain the result of the dimension estimation, we should prepare the following lemmas.
Lemma 3.1.( [1]) Suppose that the assumptions of [1] hold, the constants 0, 0 T β > > and initial value ( ) The first step will be to prove the differentiability of ( ) In what follows, we put Given ( ) ∈ , the solution ( )( ) is the linear operator where Proof.Let ( ) . First, we can prove a Lipschitz property of ( ) We now consider the difference , where , , , Taking the scalar product of each side of (3.8)-(3.9)with θ and δ , and then we have u v ( ) .
Taking the scalar product of right side of (3.16) with θ , and then we obtain , 8 , 16 , 8 .
, , , 0 The differentiability of ( ) S t is proved.The next step will be used in demonstrating the process of dimension estimation.It seems obvious that the equations (1.1)-(1.2) also can be written as Lemma 3.4.For any ( ) Proof.For any ( ) By applying the Holder inequality, Young's inequality and Poincare inequality, we deal with the terms in (3.26) by as follows ) , and substituting The proof of lemma 3.4 is completed.
Consider the first variation equation of ( P ϕ ϕ ϕ , , , 0 0 Proof.This is a direct consequence for Lemma 6. ( ) , , , n Ψ Ψ Ψ of (3.30), and we memorize that ( ) With respect to the scalar product ( ) By the Lemma 3.4, we have By the assumption (H3) in [1], the mean value theorem and the Sobolev embedding theorem  lim sup 0 The proof of theorem 3.1 is completed.
they got the existence and uniqueness of the solution by the Galerkin method and obtained the existence of the global attractor in theorem, besides, the estimation of the upper bound of Hausdorff dimension for the attractor was established.The paper is arranged as follows.In Section 2, some preliminaries and main results are stated.In Section 3, in order to acquire the result of the estimation, we show the differentiability of semigroup.Eventually, the Hausdorff and Fractal dimensions of the global attractor for the dynamics system associated with problem (1.1)-(1.5)are discussed in detail.

( 3 .
27)-(3.28) into (3.26),we obtain 3 of [2].Theorem 3.1.Let A be the global attractor of problem (1.1)-(1.5),then the Hausdorff dimension of global attractor A is less than or equal to 0 n and its fractal dimension is less than or equal to 0 2n .
R c R c R > , such that )