The Inertial Manifold for a Class of Nonlinear Higher-Order Coupled Kirchhoff Equations with Strong Linear Damping

This paper considers the long-time behavior for a system of coupled wave equations of higher-order Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of the inertial manifold while such equations satisfy the spectral interval condition.


Introduction
The concept of inertial manifold proposed by C. Foias, G. R. Sell and R. Temam [1] in 1985 is a very convenient tool to describe the long-time behavior of solutions of evolutionary equations, these inertial manifolds are smooth finite dimensional invariant Lipschitz manifolds which contain the global attractor and attract all orbits of the underlying solutions exponentially.It is closely related to infinite and finite dimensional dynamic systems, that is, the existence of inertial manifold in infinite-dimensional dynamical system is reduced to the existence of inertial manifold in finite-dimensional dynamical system.Furthermore, when the system demonstrated by restriction to the inertial manifold, it reduces to finite-dimensional ordinary differential equation, at this point, the system is called the inertial system.As in this following, the existence of such manifold relies on a spectral gap condition that turns out to be very restrictive for the applications.
It is well known that early researches on inertial manifold have yielded considerable results.In 1988, the concept of spectral barriers was utilized in the Hil-bert space to attempt to refine spectral separation condition by Constantin et al. [2], after that, the inertial manifold was constructed with using an elliptic regularization method by Fabes, Luskin, Sell in [3] (See [4] for other research results).Among then, the two well-known methods used to show the existence of inertial manifold are the Lyapunov-Perron method and the Hadamard graph transformation method.
In recent years, there have been many works which focus on using the latter method to study it.Wu Jingzhu and Lin Guoguang introduced the graph transformation method in [5] to obtain the existence of inertial manifold for a two-dimensional damped Boussinesq equation with , .
Subsequently, Xu Guigui, Wang Libo, and Lin Guoguang dealt with the existence of inertial manifold for second-order nonlinear wave equation with delays in the literature [6] under the assumption that the time lag is sufficiently small, ( ) ( ) ( ) In addition, Guo Yamei and Li Huahui obtained the existence of inertial manifold for a class of strongly dissipative nonlinear wave equation in [7]: .
Chen Ling, Wang Wei and Lin Guoguang discussed the situation of higher-order Kirchhoff equation in [8]: In this paper, basing on previous studies, the existence of the inertial manifold for nonlinear Kirchhoff type equations with higher-order strong damping is considered by using the Hadamard graph transformation method.The paper is arranged as follows.In Section 2, some notations, definitions and lemmas are given.In Section 3, in order to acquire the result of the existence of the inertial manifold, we show spectral gap condition.
where Ω 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 ( ) ( ) , , be a solution semigroup on a Banach space X, a subset X µ ⊂ is said to be an inertial manifold if it satisfies the following three properties: 1) μ is a finite-dimensional Lipschitz manifold; 2) μ is positively invariant, i.e., ( ) 3) μ attracts exponentially all orbits of solution , that is, there are constants for every x X ∈ , and the rate of decay in (2.1) is exponential, uniformly for x in bounded sets in X. property 3) implies that the inertial manifold must contain the universal attractor.
In order to describe the spectral interval condition, we firstly consider that the and operator A satisfies spectral interval condition related to F, if the point spectrum of the operator A can be divided into the following two parts 1 σ and 2 σ , where 1 = is uniformly bounded and global Lipschitz continuous functions.

Inertial Manifold
Equations (1.1)-(1.2) are equivalent to the following first-order evolution equation ( ), To determine characteristic values of operator A * , we consider the graph norm on X, which induced by the scale product where , , , ,  , , , , , , ,  U u p v q V x y z w x y z w = = represent the conjugation of , , , x y z w respectively.Moreover, the operator A * defined in (3.2) is monotone.Indeed, for ( ) therefore, ( ) is a non-negative real number.
To further determine the eigenvalues of A * , we consider the following characteristic equation .
Substituting the first and third equations of (3.7) into the second and fourth equations, thus , u v satisfy the problem of eigenvalues taking the inner product of , u v on both sides of the first and second equations of (3.8) respectively, we acquire u v to the position of , u v , for any positive integer k, the equation (3.6) has paired eigenvalues where k µ is the characteristic value of ( ) then the eigenvalues of the operator A * are all real numbers, and the corresponding characteristic functions are ( ) , , , .
For convenience, we note that for any < , and l be the Lipschitz constant of ( )( ) .
Then the operator A * satisfies the spectral interval condition of Definition 1.2.
Proof.We firstly estimate the Lipschitz property of F, from (3.1) and (3.4), we have That is 2 F l l ≤ .Next it can be known from (3.11) that k λ ± to be real num- bers if and only if ( ) By assumption ( ) 0 M s > , A * has at most number 0 N for finite real eigenvalues, and when 0 0 N = , ( ) . The eigenvalues are complex, and Re , 2 ≥ be such that (3.12) holds, decomposing the point spectrum of there is no k such that , vice versa, so 1 X and 2 X are orthogonal subspaces of X. From (2.3) and (3.14), we have 1 2 1

Re , Re
N N λ λ Re . 2 Thus, (3.12) implies that A * satisfies the spectral interval inequality (2.5), in conclusion, A * satisfies the spectral interval condition.
The proof of Theorem 3.1 is completed.Theorem 3.2 Suppose l be the Lipschitz constant of ( )( ) .
Then in either case, the operator A * satisfies the spectral interval condition The three steps to prove Theorem 3.2 are as follows: Step1 Setting ( ) ( ) In addition, if where 0 0 0 sup , Re , 2 .
For any k, there is ( ) , and according to the initial hypothesis ( ) where N P and R P are projections of X to N X and R X respectively, for briefly, (3.31) can be abbreviated as the following In the inner product of X, to prove that 1 X and 2 X are orthogonal, as long as 1 X and C X are proved to be orthogonal, i.e., , 0 , .
and Lipschitz continuous, and has a positive Lipschitz constant F l ; its operator A has several positive real part eigenvalues, and the eigenfunctions expand to the corresponding orthogonal spaces in X. Lemma 2.1 Let the operator : A X X → have countable positive real part eigenvalues whose eigenfunctions expand to the corresponding orthogonal spaces in X, and of A * are all real numbers, and we know that both { } by(3.35), then the spectral interval condition (2.5) holds if