The Exponential Attractor for a Class of Nonlinear Coupled Kirchhoff Equations with Strong Linear Damping

This paper investigates the dynamics for a class of nonlinear higher-order coupled Kirchhoff equations with strong linear damping. By means of the method proposed by Eden et al., the Lipschitz continuity and the discrete squeezing property of its solution semigroup are proved, and thus the existence of the exponential attractor is obtained.


Introduction
In 1990, an exponential attractor for a continuous map S which conducts on a compact invariant set B was defined by Eden et al.As an inertial set, and as a compact, invariant finite dimensional subset M of B containing the global attractor A , which is the  -limit set of B , and all points of B are attracted at least exponential rate.The recently developed exponential attractor theory retains many aspects of the global attractor and inertial manifold.The exponential attractor and global attractor, the main difference is that, once they are in an absorbing ball, all the solutions converge to the exponential attractor in an exponential rate, so the exponential attractor contains the global attractor, and the stable manifold convergence is only polynomial; but comparing to the inertial manifold, it also has finite dimension and attracts the solution exponentially, while the exponential attractor is not needed to have a manifold structure.The simple constructive way for exponential attractor is to restrict the inertial manifold to an absorbing set.But anyway, in general, when all the sets exist, they have the following relationship: Initially, we recall the exponential attractors of some equations that have been certified.
In cooperation with Eden, (Milani, 1992) obtained some conclusions on the existence of exponential attractors for the semi-linear damped wave equation, especially considering the case of nonlinear term in three-dimensional space: Next, (Brochet et al., 1994) considered the system of equations with simultaneous order-disordered and phase separation dynamics in 3 N  , the existences of the inertial set and the maximum attractor were proved and the upper bound of the fractal dimension of the attractor was obtained.Subsequently, the existence of exponential attractors was established by (Eden & Rakotoson, 1994), that is, there is a sufficient condition for DSP to guarantee its existence.(Eden & Kalantarov, 1996) simplified the framework by introducing a unified method to both the existence of exponential attractor by  contraction and the construction of exponential attractor by some Lipschitizianity condition of nonlinear operator.(Eden et al., 1998) had an improvement in the original construction of exponential attractor.
Authors in (Shang & Guo, 2005) considered the global fast dynamics of the generalized symmetric regularized long wave equation with damping term and got the squeezing property of nonlinear semigroup and the existence of the exponential attractor.(Lin et al., 2017) studied the global dynamics of a nonlinear generalized Kirchhoff-Boussinesq equation with damping term and proved the existence of its exponential attractor: The exponential attractors of higher-order nonlinear Kirchhoff equation were analyzed by (Chen et al., 2016): Inspired by the above, this article arranges as follows.In Part 2, some of the main preliminaries are stated, and in Part 3, the Lipschitz continuity and discrete squeezing property of semigroup are acquired, thereby exponential attractor is established.
where  is a bounded domain in

Preliminaries
For convenience, we need the following notations in subsequent article.Considering a family of Hilbert spaces   , whose inner product and norm are given by 22 ( , ) ( , ) We make the following hypotheses: , ( ) 0.
Definition 1 (Eden et al., 1994) A compact set  , where M has finite fractal dimension; 4) There exist universal constants 12 , cc , such that for every uB  ,for every natural number t , 2 1 ( ( ) , ) Definition 2 (Eden et al., 1994) A solution semigroup   0 () St  is said to satisfy the discrete squeezing property (DSP)   if there exists * 0 t  such that the map ** () S S t  satisfies: there exists an orthogonal projection P of finite rank N such that, for every u and v in B , either Where Definition 3 (Eden et al., 1994) We say () for all , x y B  .Here () Lt does not depend on x and y .

The Existence of Exponential Attractor
In this section, we prove equations (1.1)-(1.2) admit an exponential attractor, we verify the Lipschitz continuity and the discrete squeezing property of the dynamical system () St in 0 E .
First, we introduce A   , since A is self-adjoint, positive operator and has a compact inverse.Let   1 be the sequence of the eigenvalues and   1 the corresponding sequence of eigenvectors, 12 (0 , ) and by the definition of projection ), ( , , , ) For each 0 ( , , , ) , , , then the norm derived from () Gz is equivalent to the norm on 0 E .Namely, 0 ( , , , ) , then the norm derived from () Fz is equivalent to the norm on 0 E .Namely, 0 ( , , , ) Proof. 1) According to (H1), we can get the conclusion easily.
2) Due to , applying Holder and Young inequality, and noticing the result of the projection, we get Lemma 2 Suppose (H1)-(H2) hold, , uv and , uv are two solutions of problem (1.1)-(1.5),let k b a  , then we have 00 22 0, ( ) (0) Where c is a constant depending only on the data ) Taking the inner product of (3.6) with t  and t  respectively, and adding them, we obtain Then by the assumption (H2), there exist constants 12 , KK, such that , By formula (3.1) Using Gronwall inequality Further, by (3.3) the equivalence of norm 00 22 ( ) (0) Lemma 2 is proved completely.
Bring (3.11) into (3.10),we obtain Taking the inner product of (3.13) with tt u and tt v respectively, we obtain From Theorem 2.1 and Theorem 2.2 of (Lin & Hu, 2017) therefore, Lemma 4 is certified by Gronwall inequality.
Lemma 5 For 0 T , the map ( , ) ( ) t z S t z is Lipschitz continuous on   0,TB  .

Proof. For
The first term on the right side of the above formula is easily handled by Lemma 2, for the second item, by virtue of Lemma 4, we obtain Proof of Theorem 1 is completed.

Conclusions
In this paper, we study a class of high-order Kirchhoff-type equations.By using the method proposed by Eden et al. and combining our assumptions given in advance, we obtain the Lipschitz property and the discrete squeezing property, which prove its the existence of exponential attractor.Among them, we have maximum and minimum values in the bounded closed region according to the continuous function of mathematical analysis, we have made a limitation on () Ms, which needs further improvement.In later studies, I hope that I can deeply explore its greater possibilities.