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  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 can contact finite dimensional dynamical system and infinite dimensional dynamical system, many scholars have done a great deal of research, we could refer to (  -  ).
Guigui Xu, Libo Wang and Guoguang Lin  studied global attractor and inertial manifold for the strongly damped wave equations
The assumption of satisfies the following conditions:
(H2) There is a positive constant , such that .
Under these reasonable assumptions, according to Hadamard’s graph transformation method, the existence of the inertial manifolds for the equation is obtained.
Zhijian Yang and Zhiming Liu  studied the existence of exponential attractor for the Kirchhoff equations with strong nonlinear strongly dissipation and supercritical nonlinearity
The main result was that the nonlinearity is of supercritical growth and they established an exponential attractor in natural energy space by using a new method based on the weak quasi-stability estimates.
Ruijin Lou, Penghui Lv, Guoguang Lin  studied the exponential attractor and inertial manifold of a higher-order kirchhoff equations
where is finite region of , is smooth boundary, and is initial value, is strongly damped term, is stress term, is nonlinear source term.
On the basis of reference , the stress term is extended to , 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 (  -  ).
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:
where , , is a bounded domain with a smooth boundary , is a real function, denotes strong damping term, is nonlinear source term, denotes the external force term. The assumption of and as follow:
where are constant, is the first eigenvalue of with homogeneous Dirichlet boundary conditions on .
For convenience, define the following spaces and notations , , , , , , ( ), . and represent the inner product and norms of H respectively, i.e.
, , , , .
2. Exponential Attractors
For brevity, define the inner product and norms as follow: ,
Let , , , we can get the Equation (1.1) is equivalent to the following evolution equation
Then, we will use the following notations. Let are two Hilbert spaces, we have ↪ with dense and continuous injection, and ↪ is compact. Let is a map from into .
In the following definitions, .
Definition 2.1.  The semigroup possesses a -compact attractor , If it exists a compact set , attracts all bounded subsets of , and under the function of , is an invariant set, i.e. .
Definition 2.2.  If and 1) ; 2) has finite fractal dimension, ; 3) there exist universal constants , such that , where is the positive invariant set of , the compact set is called a -exponential attractor for the system .
Definition 2.3.  if there exists limited function , such that
Then the semigroup is Lipschitz continuous in .
Definition 2.4.  If and exists an orthogonal projection of rank such that for every ,
Then is said to satisfy the discrete squeezing property, where .
Theorem 2.1.  Assume that 1) possesses a -compact attractor ; 2) it exists a positive invariant compact set of ; 3) is a Lipschitz continuous map with Lipschitz constant l on , and satisfies the discrete squeezing property on . Then has a -exponential attractor , and on , and , . Moreover, the fractal dimension of satisfies , , where is the smallest N which make the discrete squeezing property established.
Proposition 2.1.  There is such that is
the positive invariant set of in , and attracts all bounded subsets of , where is a closed bounded absorbing set for in .
Theorem 2.2.  Assuming the stress term and the nonlinear term satisfies the condition (A1)-(A2), , , then problem (1.1)-(1.3) admits a unique solution . This solution possesses the following properties:
We denote the solution in Theorem 2.1 by . Then composes a continuous semigroup in . According to Theorem 2.1, we have the ball
are absorbing sets of in and respectively. From Proposition 2.1
is a positive invariant compact set of in , and absorbs all of the bounded subsets in . According to reference  and theorem 2.1, we can get the semigroup possesses -compact global attractor
, where the bar means the closure in , and is bounded in .
Lemma 2.1. For any ,
Proof. By (2.1) and (2.2), we have
By using Holder’s inequality, Young’s inequality and Poincare’s inequality and the condition (A2), we have,
Substitute inequality (2.12)-(2.13) into Equation (2.11), we get
According to the assumption, we can get , , . Let , , so we can get
The Lemma 2.1 is proved. Then we prove the Lipschitz property and the discrete squeezing property of .
Set , where ; and , where ; let , where , , , then satisfies
Lemma 2.2. (Lipschitz property). For and ,
Proof. Taking the inner product of the Equation (2.16) with in , we can get
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
Substitute inequality (2.20)-(2.21) into equation (2.19), we get
We can get
According to Gronwall’s inequality, we have
where . Therefore, we get
The Lemma 2.2 is proved. n
Now, we define the operator : , the domain of definition is , obviously, is an unbounded self-adjoint closed positive operator, and is compact, we find by elementary spectral theory the existence of an orthonormal basis of H consisting of eigenvectors of , such that:
For a given integer n, we denote by the orthogonal projection of onto the space spanned by i.e. , let . Then we have
Lemma 2.3. For any , , , Let
then we have
Proof. Taking projection operator in (2.16), we have
Taking the inner product in (2.31) with , we get
According to (A1) and Young inequality, we have
Together with (2.32)-(2.33) and Lemma 2.2, it follows
By using Gronwall’s inequality, we get
The Lemma 2.3 is proved. n
Lemma 2.4. (Discrete squeezing property). For any , , if
Proof. If , then
Let be large enough,
Also let be large enough, we get
Substitute inequality (2.39)-(2.40) into Equation (2.38), we get
The Lemma 2.4 is proved. n
Theorem 2.3. Let (A1), (A2) be in force, assume that , , ( ), then the semigroup determined by (1.1)-(1.3) possesses an -exponential attractor on B,
The fractal dimension of satisfies
Proof. According to Theorem 2.1, Lemma 2.2 and Lemma 2.4, Theorem 2.2 is easily proven. n
3. Inertial Manifolds
Next, we will prove the existence of inertial manifolds when N is large enough by using graph norm transformation method.
Definition 3.1.  Assume is a solution semigroup of Banach space , then a family of inertial manifolds is a subset of and satisfies the following three properties:
1) is finite dimensional Lipschitz manifold of ;
2) is positively invariant for the semigroup , i.e. , , ;
3) attracts exponentially all the orbits of the solution, i.e. , for , , such that
Lemma 3.1. Let be an operator and assume that satisfies the Lipschitz condition
The operator is said to satisfy the spectral gap condition relative to F, if the point spectrum of the operator can be divided into two parts and , of which is finite, and we have
and the orthogonal decomposition
Then and are both continuous orthogonal projections . The Lemma 3.1 is proved.
Lemma 3.2. Let the eigenvalues is non-decreasing, and for , there exists , such that and are consecutive adjacent values.
Lemma 3.3. The function satisfies which is uniformly bounded and globally Lipschitz continuous, and l is the Lipschitz coefficient.
Proof. For , we have
where , From the hypothesis (A1) and the differential mean value theorem, we know
Let , is the Lipschitz coefficient. n
Then we prove the existence of a family of the inertial manifold of this equation, Equation (1.1) is equivalent to the following first-order evolution equation:
We consider in the usual graph norm, induced by the scalar product
where , , and respectively denote the conjugation of y and z, and , . Moreover, the operator is monotone, indeed, for , we have
so that is a Monotonically increasing operator and is real and nonnegative. To determine the eigenvalues of , we observe that the eigenvalue equation
is equivalent to the system
Thus, we can get the eigenvalue problem
Using with the first formula of (3.13) to take the inner product, and bring to the position of u, we can get
Regarding Equation (3.14) as a quadratic equation of one variable with respect to , for and let , , the corresponding eigenvalues of equation (3.11) are as follows:
where is the eigenvalue of in , and . Because of is large enough, the eigenvalue of are all positive and real numbers, the corresponding eigenvalues have the form
For formula (3.15), for the convenience of later use, define the following formula
Next, it will be proved that the eigenvalue of the operator satisfies the spectral interval condition.
Theorem 3.1 let l is the Lipschitz constant of , assume , if is large enough, when , the following inequality holds
Then, the operator satisfies the spectral gap condition of Lemma 3.1.
Proof. Because of all the eigenvalues of the operator are positive real numbers, and the sequence and are monotonically
increasing. The theorem is proved in four steps below.
step 1 Since is a non-decreasing sequence, according to Lemma 3.2, given N, so that and are consecutive adjacent eigenvalues, the eigenvalues of the operator are decomposed into and , where is the finite parts, which are expressed as follows
step 2 The corresponding is decomposed into
We aim at madding two orthogonal subspaces of and verifying the spectral gap condition (3.4) is true when . Therefore, we further decompose , i.e.
And set . Note that and are finite dimensional, that , , and that the reason why is not orthogonal to is that, while it is orthogonal to , is not orthogonal to . We now introduce two functions and , defined by
where , and are respectively the conjugates of . We now show that and are positive definite. For , we have
When is large enough, we conclude that , i.e. is positive definite. Similarly, for , we have
When is large enough, we conclude that , i.e. is positive definite.
Thus and define a scalar product, respectively on and , and we can define an equivalent scalar product in , by
where and are respectively the projections of and . Rewrite (3.29) as follows
We proceed then to show that the subspaces and defined in (3.21), (3.22) are orthogonal with respect to the scalar product (3.29). In fact, it is sufficient to show that is orthogonal to , in turn, this reduces to showing that if and . Recalling (3.27) and (3.28), we immediately compute that
According to (3.15), we have
step 3 Further, we estimate the Lipschitz constant of , according to Lemma 3.3 we can get is uniformly bounded and globally Lipschitz continuous. For , , we have
Given , we have
Thus, we have
step 4 Now, we will show the spectral gap condition (3.4) holds.
Since , then
There exists , such that for , . We can get
According to assumption (A2), we can easily see that
Then according to (3.18) and (3.37)-(3.40), we have
The Theorem 3.1 is proved.
Theorem 3.2. Under the conclusion of Theorem 3.1, the problem (1.1)-(1.3) exists a family of inertial manifolds in
where defined in (3.21)-(3.22), and is Lipschitz continuous function. n
 Huy, N.T. (2012) Inertial Manifolds for Semi-Linear Parabolic Equations in Admissible Space. Journal of Mathematical Analysis and Applications, 386, 894-909.
 Zhong, Y.S. and Zhong, C.K. (2012) Exponential Attractors for Semigroups in Banach Spaces. Nonlinear Analysis: Theory, Methods & Applications, 75, 1799-1809.
 Yang, Z. and Liu, Z. (2015) Exponential Attractor for the Kirchhoff Equations with Strong Nonlinear Damping and Supercritical Nonlinear. Applied Mathematics Letters, 46, 127-132.
 Fan, X.M. and Yang, H. (2010) Exponential Attractor and Its Fractal Dimension for a Second Order Lattice Dynamical System. Journal of Mathematical Analysis and Applications, 367, 350-359.
 Lou, R.J., Lv, P.H. and Lin, G.G. (2016) Exponential Attractors and Inertial Manifolds for a Class of Generalized Nonlinear Kirchhoff-Sine-Gordon Equation. Journal of Advances in Mathematics, 12, 6361-6375.
 Lin, G.G. and Yang, L.J. (2021) Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations. Advances in Pure Mathematics, 11, 317-333.