A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation

In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.


Introduction
In this paper, we considers the initial-boundary value problem of the following higher-order nonlinear Kirchhoff- Where Ω is a bounded domain in ( 1)  In 1883, Gustav Robert Kirchhoff [1], a German physicist, first introduced the following equation when he studied the transverse vibration of stretched strings Where h is the cross-sectional area of the string, E is young coefficient, 0 P is the initial axial tension, L is the length of the string, ρ is the mass density of the string, and ( , ) u u x t = is the lateral displacement in the spatial axial coordinate. Over the past a hundred years, with the development of science and technology, Kirchhoff equation has been paid more and more attention by scholars. The application field of Kirchhoff equation is also expanding, and its expression is constantly extended. A series of mathematical theories and research results have been obtained, such as the existence and uniqueness of global solution, the decay of solution, the existence of random attractor, and the index Numerical attractor, global attractor and its dimension estimation, inertial manifold, etc [2][3][4][5][6].
When m = 1, q = 2, the stress term is 2 2 (1 ( ) ) u t + ∇ , and a nonlinear nonlocal source term is added to equation (1), Mitsuhiro Nakao [7] studied the existence of attractors and some absorption properties in the local sense for this class of Kirchhoff type quasilinear wave equation with standard dissipation term t u . Then, Zaiyun Zhang [8] and others studied the initial boundary value problem of nonlinear dissipative Kirchhoff   studied the well posedness and long-time behavior of the solution of the initial boundary value problem of the equation By assuming the Kirchhoff term, he proved the existence and uniqueness of weak solution and the existence of a finite dimensional global attractor in the natural energy space with partial strong topology, and further proved that the attractor is strong under non supercritical conditions.
Recently, on the basis of chueshov Igor [9], Guoguang Lin [10] and others studied the long-term behavior of the initial boundary value problem for a class of nonlinear strongly damped higher order Kirchhoff type equation with They obtain the existence and uniqueness of the solution and the global attractor, and consider the dimension of the global attractor and the upper bound estimation of the dimension. For more related research results on the Kirchhoff equation, please refer to [11][12][13][14][15].
In this paper, based on the long-time behavior of solutions of some nonlinear Kirchhoff type equations with initial boundary value problems, a class of higher order Kirchhoff type equations with the highest order term L Ω , there will be a bottleneck when using uniform prior estimation and Galerkin finite element method to prove the existence and uniqueness of the global solution of the equation, so it is impossible to continue the follow-up work. We get a relation between m and q in Banach space ( ) q L Ω by the theory of Sobolev space. Therefore, we overcome this problem successfully and get more extensive research methods and theoretical results.

Preliminaries
For brevity, we used the follow abbreviation:

The Existence of a Family of Global Attractors
Thus, there exists a non-negative constant k R and By using ö H lder Inequality, Young inequality and é Poincar inequality, the following are obtained by dealing with the following items in formula (4) By assuming (A1) and using Young inequality and é Poincar inequality to deal with the strong damping term, we can obtain It can be obtained from hypothesis (A2) ( ) Using Schwarz inequality and Young inequality to deal with the external force term, we obtain Substituting (5) -(9) into (4), we receive According to hypothesis (A1), we have 2 0 1 1 By using Gronwall inequality, we obtain ( ) So, we have ( ) Thus, there exist a non-negative constant Lemma 1 is proved Theorem 1 Under the hypotheses of Lemma 1, and ( ) , Proof. Existence: the existence of global solution is proved by Galerkin method Step 1: construct approximate solution Let ( ) , 0,1, 2, 3, , , is determined by the following system of differential equations When s → +∞ , we can obtain 0 1 It can be seen that the priori estimates of the solution of lemma 1 in formula (19) and (20) Because s u u → weakly * converges in As a result, In particular, 0 weakly converges in k E . For any j and l → +∞ , we can get Because of the arbitrariness of j w , we have for any B 1 0 Therefore, the existence is proved. Next, we prove the uniqueness of the solution. Let * * , u v be two solutions of the system of equations, let By using Sobolev embedding theorem, we can obtain that, Similarly, there are Combined with formula (24) -(31), it is concluded that Further, due to so, is a bounded absorbing set in k E and satisfies the following conditions: From lemma 1, we can obtain that Furthermore, for any 0 0 ( , ) Therefore, is a bounded absorbing set in semigroup ( ) S t . According to the rellich kondrachov compact embedding theorem, if k E is compactly embedded in 0 E , then the bounded set in k E is the compact set in 0 E . Therefore, the solution semigroup ( ) S t is a completely continuous operator, thus the global attractor family k A of solution semigroup ( ) S t is obtained. Where The prove is completed.

Dimension Estimation
In this part, we first linearize the equation into a first-order variational equation and prove that the solution semigroup ( ) s t is échet Fr differentiable on k E . Furthermore, we prove the decay of the volume element of the linearization problem. Finally, we estimate the upper bound of the Hausdorff dimension and fractal dimension of k A The initial boundary value problem (1) -(3) is linearized and rewritten into a first order variational equation We define 1 S t is Lipschitz continuous on the bounded set of k E , i.e. where By using the inner product in k E of Equation (47) with φ , we obtain Further, let where 2