The concept of inertial manifold proposed by C. Foias, G. R. Sell and R. Temam  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 Hilbert space to attempt to refine spectral separation condition by Constantin et al.  , after that, the inertial manifold was constructed with using an elliptic regularization method by Fabes, Luskin, Sell in  (See  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  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  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  :
Chen Ling, Wang Wei and Lin Guoguang discussed the situation of higher-order Kirchhoff equation in  :
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.
is a bounded domain in
with smooth boundary
is real number and
is positive integer,
is a nonnegative
are nonlinear terms and external force terms respectively.
For convenience, we need the following notations in subsequent article. Considering a family of Hilbert spaces
, whose inner product and norm are given by
Definition 2.1  Let
be a solution semigroup on a Banach
space X, a subset
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.,
, for all
3) μ attracts exponentially all orbits of solution , that is, there are constants
, and the rate of decay in (2.1) is exponential, uniformly for
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 nonlinear term
is globally bounded and Lipschitz continuous, and has a positive Lipschitz constant
; 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
have countable positive real part eigenvalues whose eigenfunctions expand to the corresponding orthogonal spaces in X, and
satisfies the Lipschitz condition:
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
hold with continuous orthogonal projection
is uniformly bounded and global Lipschitz continuous functions.
Lemma 2.3 Let eigenvalues
be arranged in non-decreasing order, then for
, there is
are adjacent values.
3. Inertial Manifold
Equations (1.1)-(1.2) are equivalent to the following first-order evolution equation
To determine characteristic values of operator
, we consider the graph norm on X, which induced by the scale product
represent the conjugation of
respectively. Moreover, the operator
defined in (3.2) is monotone. Indeed, for
is a non-negative real number.
To further determine the eigenvalues of
, we consider the following characteristic equation
Substituting the first and third equations of (3.7) into the second and fourth equations, thus
satisfy the problem of eigenvalues
taking the inner product of
on both sides of the first and second equations of (3.8) respectively, we acquire
that is to say
(3.10) is a quadratic equation about
to the position of
, for any positive integer k, the equation (3.6) has paired eigenvalues
is the characteristic value of
then the eigenvalues of the operator
are all real numbers, and the corresponding characteristic functions are
For convenience, we note that for any
Theorem 3.1 Suppose
be the Lipschitz constant of
in (3.1) , set
be so large such that if
Then the operator
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
. Next it can be known from (3.11) that
to be real numbers if and only if
. By assumption
has at most number
for finite real eigenvalues, and when
. The eigenvalues are complex, and
therefore, there exists
be such that (3.12) holds, decomposing the point spectrum of
meanwhile, define the corresponding subspaces of X
there is no k such that
, i.e., it is impossible to have
, vice versa, so
are orthogonal subspaces of X. From (2.3) and (3.14), we have
Thus, (3.12) implies that
satisfies the spectral interval inequality (2.5), in conclusion,
satisfies the spectral interval condition.
The proof of Theorem 3.1 is completed.
Theorem 3.2 Suppose
be the Lipschitz constant of
in (3.1), assume
be large enough, when
, the following inequalities hold, either
Then in either case, the operator
satisfies the spectral interval condition (2.5).
Proof. Due to
, the eigenvalues of
are all real numbers, and we know that both
are monotonically increasing sequences.
The three steps to prove Theorem 3.2 are as follows:
; and if
In addition, if
Step 2 Consider the corresponding decomposition of X
the equivalent inner product
on X will be given below so that
are orthogonal. Given
Now we introduce two functions
, defined as
For any k, there is
, and according to the initial hypothesis
, that is
is positive definite.
is positive definite.
Specify the inner product of X:
are projections of X to
respectively, for briefly, (3.31) can be abbreviated as the following
In the inner product of X, to prove that
are orthogonal, as long as
are proved to be orthogonal, i.e.,
according to (3.10)
thus, (3.33) is equivalent to
Step3 The orthogonal decomposition (2.6) has now been established. Let us prove that
satisfies the spectral interval condition (2.5) and its equivalent norm on X is shown in (3.31), for this, we must estimate Lipschitz constant
of F in (2.2).
is Lipschitz continuous. Assume
be the orthogonal maps of
are their corresponding mappings on
, from (3.29) and (3.30), for
, we get
by (3.35), then the spectral interval condition (2.5) holds if
Recalling (3.22), we have
For formula (3.38), in fact, setting
Consequently, (3.38) is obtained.
From the condition (3.19), it can be determined that
such that for all
, and with (3.37)
this shows that (3.36) is established by the conditions (3.18), (3.37), and (3.40), that the Theorem 3.2 is certified completely under the previous hypothesis.
At this point, we continue to use the latter hypothesis to prove, setting
then (3.37) is equivalent to
then (3.41) means
from (3.43), we easily get
to be specific
Under the latter assumption, Theorem 3.2 is proved completely.
Theorem 3.3 In the conclusions of Theorem 3.1 and Theorem 3.2, initial boundary value problems (1.1)-(1.5) admits an inertial manifold
in X of the form
is Lipschitz continuous with the Lipschitz constant
represents the diagram of
The authors express their sincere thanks to the anonymous referee for his/her careful reading of the paper, giving valuable suggestions and comments, which have greatly improved the presentation of this paper.