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 0k B 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 k A in the phase space 2m k k k E H H + = × . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on k E . Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor k A was obtained.


Introduction
The objective of this paper is to study the following initial boundary value problem of the generalized Kirchhoff equation In 1883, German physicist G. Kirchhoff [1] first introduced the following model to study the free vibration of elastic strings ( ) ( ) where the time variable is t, the elastic modulus is E, h is the cross-sectional area, L is the length of the string, ρ is the mass density, 0 P is the initial axial tension, δ is the resistance coefficient, f is the external force term, the lateral displacement at the space coordinate x and the time t.
Since the 1980s, with the progress of science and technology and the conti- By assuming the nonlinear source terms g(u), the author verifies the appropriateness of the solution and proves the existence of the global attractor.
Recently, Lin Guoguang and Guan Liping [11]  In this paper, on the basis of literature [11], the rigid term For convenience, define the following spaces and notations Assume that the nonlinear source term g(u) in Equation (1.1) satisfies the following conditions The Kirchhoff-type stress term satisfies the following conditions µ µ µ are constant, and λ is the first eigenvalue of −∆ with homogeneous Dirichlet boundary conditions on Ω .

A Priori Estimate of Smooth Solution
there's a non-negative real number Proof. Set with v in H, we obtain The following estimation can be obtained from hypothesis (A2) By using the weighted Young's inequality, we obtain By using the Gronwall's inequality, we get So, there are constants The Lemma 1 is proved.
Lemma 2. Assume that the nonlinear terms g(u), M(s) satisfies assumptions there are non-negative real number According to hypothesis (A5), and use a proof method similar to lemma 1, we can get By using Poincare's inequality and Young's inequality, we have Obviously, there is a non-negative ε , such that The Lemma 2 is proved. 2) Prior estimate.

Existence and Uniqueness of Solutions
According to the conclusion and proof method of lemma 1, where R is a constant independent of k. According to lemma2, we get , , Because of ( ) is satisfied for all j, so that existence can be proved.
Then prove the uniqueness of the solution.
, then the uniqueness of solutions is proved.
The theorem 1 is proved completely.

The Existence of the Family of Global Attractor
Theorem 2 [16] Assume that E is a Banach space, operator on E, and ( ) : where I is unit operator, suppose ( ) S t satisfies the following conditions:

1) Semigroup S(t) is uniformly bounded in E, that is for all
is completely continuous operator.
Thus there is a compact global attractor 0 A for the semigroup operator ( ) S t .
If the Banach space E is changed to Hilbert space k E in theorem 2, the existence theorem of the family of the global attractors can be obtained. , , fies the following conditions:  .

Estimation of the Dimension of the Family of Global Attractors
Let's consider the linearization problem of (1.1)-(1.3) Theorem 4. The Frechet derivative of mapping ( ) : k , c is a constant. We can get the Lipschitz property of S(t) on the bounded set k E , that is Take the inner product of (5.5) with ( ) The theorem 4 is proved.