In this paper, we concerned the equation:
where Ω is a bounded domain in Rn with a smooth boundary , is a constant and is a given source term. Moreover, is a scalar function. Then the assumptions on M and will be specified later.
Nowadays, the study on the complexity of the space-time of high dimensional and infinite dimensional dynamical systems has gradually become the focus of nonlinear scientific research. In recent years, the inertial manifold has been found in the researches of the long time behavior of the solution and the attractor structure. The inertial manifold is a tool to describe the interaction between the low frequency components and the high frequency components  . When the flow has an inertial manifold, its high frequency description depends on the low frequency, and it contains attractors and exponentially attracts solution of the track, which realizes that the infinite dimensional dynamical system is reduced to a finite dimensional dynamical systems of the finite dimensional invariable Lipschitz manifold. Therefore, the inertial manifold is a powerful tool to study the long-time behavior of nonlinear dissipative systems and expose the real or seemingly chaotic structure of nonlinear dynamics.
In addition, the study of inertial manifold is of great significance. The central idea of the methods that people use to solve practical problems such as Galerkin method, Cellular automaton and Coupled map, are to discuss the infinite dimensional problem into a finite dimensional problem. So, the inertial manifold is of great significance to the development of nonlinear science.
In 1988, the concept of inertial manifold was first proposed in the study of infinite dimensional dynamical system by R. Temam, C. Foias and Sell G.R.  . They considered the equation as following:
where Au is a linear unbounded self-adjoint operator on H with domain dense in H.
In 2010, Guoguang Lin and Jingzhu Wu  studied the existence of the inertial manifold of Boussinesq equation:
In 2016, Ling Chen, Wei Wang and Guoguang Lin  established the exponential attractors and inertial manifolds of the higher-order Kirchhoff-type equation:
There are many researches on inertial manifolds for nonlinear wave equations (see   ). Concerning the inertial manifold, many difficulties are solved. So we take advantage of Hadamard’s graph transformation method in this paper.
The paper is arranged as follows. In Section 2, some assumptions, notations and lemmas are stated. In Section 3, the existence of the inertial manifold is established.
For convenience, we first introduce the following notations:
denotes different positive constants, and are the inner product and norm of , is the norm of .
Next, we give some assumptions and definition needed in the proof of our results.
Definition 2.1.  Let be an operator and assume that satisfies the Lipschitz condition
The operator A is called satisfy the spectral gap condition relative to F, if the point spectrum of the operator A can be divided into two parts and , of which is finite, and such that, if
and the orthogonal decomposition holds with continuous orthogonal projections .
Lemma 2.1.  Let the eigenvalues be arranged in nondecreasing order. For all , there exists such that and are consecutive.
3. The Inertial Manifold
Equation (1.1) is equivalent to the following one order evolution equation:
We consider the usual graph norm in X, as follows
where respectively represent the conjugation of . Evidently, the operator A is monotone, for , we obtain
So, is a nonnegative and real number.
In order to determine the eigenvalues of A, we consider the eigenvalues equation:
Substitute (3.6), (3.7) into (3.8), (3.9), we obtain
Replacing with , taking inner product with the Equations (3.9), (3.10), and adding them together, we have
(3.11) is regard as a quadratic equation with one unknown about , so we get
for , we have
, , . (3.13) and is non-derogatory. If , because of , then , we can get the eigenvalues of A are all positive and real numbers.
The corresponding eigenfunction is as follows
Lemma 3.1. is uniformly bounded and globally Lipschitz continuous.
Proof. , by , we have
Let , then is Lipschitz coefficient of .
Theorem 3.1. is Lipschitz constant of , when , set , such that , we obtain
where . By and Lemma 3.1, the operator A satisfies the spectral gap condition of (2.4).
Proof. when , the eigenvalues of A are all positive and real
numbers, meanwhile and are increasing order.
Next, we divided the whole process of proof into four steps.
Step 1 By Lemma 2.1, since is nondecreasing order, so there exists N, such that and are continuous. Then the eigenvalues of A are separate as
Step 2 The corresponding X is decomposed into
We aim at madding two orthogonal subspaces of X and verifying the spectral gap condition (2.4) is true when . Therefore, we further decompose , where
Set , in order to verify the and are orthogonal, we need to introduce two functions , .
where are defined before.
Let , by , then
Since for , therefore , for , then is positive definite.
Similarly, for , we have
So, for , the is also positive definite.
Next, we need to define a scale product in X
where and are projection: respectively, for convenience, we rewrite (3.24) as follows
We will proof that two subspaces X1 and X2 in (3.18) are orthogonal. In fact, we only need to show XN and XC are orthogonal, that is
By (3.20), (3.25), we have
By (3.12), we can get , therefore
Step 3 Further, we estimate the Lipschitz constant of (3.2). According to Lemma 3.1, is Lipschitz continuous with Lipschitz constant . Let is orthogonal projection. From (3.22), (3.23) and (3.24), we have
Given , we have
So, we obtain
Step 4 Now, we will show the spectral gap condition (2.4) holds.
Since , then
Let , for , then , we can obtain
From , we can easily obtain
Then, according to (3.16), (3.31), (3.32) and (3.34), we have
Therefore, Theorem 3.1 is true.
Theorem 3.2. Under the condition of Theorem 3.1, the problem (1.1)-(1.5) exist an inertial manifold in X,
where defined in (3.18) and is a Lipschitz continuous function.
The authors would like to thank for the anonymous referees for their valuable comments and suggestions sincerely. These contributions increase the value of the paper.
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