A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.


Introduction
In the study of dynamic behavior for a long time in infinite dimensional dynamical system, the exponential attractors and inertial manifolds play a very important role. In 1994, Foias [1] puts forward the concept of exponential attractor, it is a positive invariant compact set which has finite fractal dimension and attracts solution orbits at an exponential rate. Inertial manifold is finite dimensional invariant smooth manifolds that contain the global attractor and attract all solution orbits at an exponential rate, their corresponding inertial manifold forms are powerful tools which could study the property of finite dynamical system about the dissipative evolution equation. Under the restriction of inertial manifold, a infinite dimension dynamical system could be transformed to finite dimension, therefore, the inertial manifolds become an important bridge which The assumption of ( ) g u satisfies the following conditions: where Ω is finite region of n R , ∂Ω is smooth boundary, On the basis of reference [11], the stress term 2 m D u is extended to p m p D u , this paper studied the long-time dynamic behavior of a class of generalized Kirchhoff equation. Firstly, the existence of the exponential attractor of this equation is proved. Furthermore, the existence of a family of inertial manifold is proved by using Hadamard's graph transformation method, more relevant research can be referred to ([12]- [17]). Journal of Applied Mathematics and Physics In this paper, we study the existence of exponential attractors and a family of the inertial manifolds for a class of generalized Kirchhoff-type equation with damping term: The assumption of ( ) M s and ( ) g u as follow: neous Dirichlet boundary conditions on Ω .
For convenience, define the following spaces and notations ( )

Exponential Attractors
For brevity, define the inner product and norms as follow: .
Then the semigroup ( ) Then ( ) S t is said to satisfy the discrete squeezing property, where N which make the discrete squeezing property established.
Proposition 2.1. [15] There is ( ) . This solution possesses the following properties: are absorbing sets of ( ) is a positive invariant compact set of ( ) By using Holder's inequality, Young's inequality and Poincare's inequality and the condition (A2), we have, According to the assumption, we can get The Lemma 2.1 is proved. Then we prove the Lipschitz property and the discrete squeezing property of ( ) Proof. Taking the inner product of the Equation (2.16) with ( ) Similar to Lemma 2.1, we have By using the condition (A1) Young's inequality Poincare's inequality and differential mean value theorem, we get The Lemma 2.2 is proved.  Now, we define the operator −∆ :  , .
Let * τ be large enough,

Inertial Manifolds
Next, we will prove the existence of inertial manifolds when N is large enough by using graph norm transformation method. µ is a subset of k E and satisfies the following three properties: 3) k µ attracts exponentially all the orbits of the solution, i.e.
be an operator and assume that ( ) The operator Λ is said to satisfy the spectral gap condition relative to F, if the point spectrum of the operator Λ can be divided into two parts 1 σ and 2 σ , of which 1 σ is finite, and we have Then 2 1 4 , F l Λ − Λ > (3.4) and the orthogonal decomposition is equivalent to the system , .
For formula (3.15), for the convenience of later use, define the following for- Next, it will be proved that the eigenvalue of the operator Λ satisfies the spectral interval condition.
The Theorem 3.1 is proved.