This paper mainly deals with existence of inertial manifolds for a class of higher-order coupled Kirchhoff-type equations:
is an integer constant,
is a bounded domain of
with a smooth Dirichlet boundary
are the initial value.
are the unit outward normal on
is a nonnegative
are strongly damping,
are nonlinear source terms,
are given forcing functions.
It is significant to establish inertial manifolds for the study of the long-time behavior of infinite dimensional dynamical systems, because it is an important bridge between infinite-dimensional dynamic system and finite-dimensional dynamical system.
To better carry out our work, let’s recall some results regarding wave equations.
Jingzhu Wu and Guoguang Lin  studied the following two-dimensional strong damping Boussinesq equation while
. They obtained result that is existence of inertial manifolds.
Guigui Xu, Libo Wang and Guoguang Lin  investigated the strongly damped wave equation:
They gave some assumptions for the nonlinearity term
to satisfy the following inequalities:
(A2) There is positive constant
According to the above assumptions, they proved the inertial manifolds by using the Hadamard’s graph transformation method.
Ruijin Lou, Penhui Lv, Guoguang Lin  considered a class of generalized nonlinear Kirchhoff-Sine-Gordon equation:
Under some reasonable assumptions, they obtained some results that are squeezing property of the nonlinear semigroup associated with this equation and the existence of exponential attractors and inertial manifolds.
Lin Chen, Wei Wang and Guoguang Lin  studied higher-order Kirchhoff-type equation with nonlinear strong dissipation in n dimensional space:
For the above equation, they made some suitable assumptions about
to get existence of exponential attractors and inertial manifolds.
In this article, we first take advantage of Hadamard’s graph to transform equations (1.1)-(1.2) into an equivalent one-order system of form. Then, we proved the inertial manifolds by using spectral gap condition.
We denote the some simple symbols,
and (,) stand for norm and inner product of H and
are various positive constants.
Next, we give some assumptions needed for problem (1.1)-(1.6).
Then, we give the basic concepts below.
Definition 2.1.  An inertial manifold μ is a finite-dimensional manifold enjoying the following three properties:
1) μ is Lipschitz,
2) μ is positively invariant for the semi-group
(3) μ attracts exponentially all the orbits of the solution.
Definition 2.2.  Let
be an operator and assume that
satisfies the Lipschitz condition:
. The operator A is said to satisfy the spectral gap condition relative to F, if the point spectrum of the operator A can be divided into two parts
, 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 is
3. Inertial Manifold
In this section, we use the Hadamard’s graph transformation method to prove the existence of inertial manifolds of problem (1.1)-(1.6) when N is sufficiently large.
Equations (1.1)-(1.2) are equivalent to the following one order evolution equation:
In X, we denote the usual graph norm, which is introduced by the scalar product as the following form
denote the conjugation of
, we have
Therefore, the operator H in (3.2) is monotone, and
is a nonnegative and real number.
To obtain the eigenvalues of H, we consider the following eigenvalue equation:
The first two equations in (3.8) are brought into the last two equations in (3.8) respectively, we get
replace u and v in (3.9) respectively. And then taking
inner product respectively, we obtain
Summing up (3.10) and (3.11), we get
When (3.12) is considered an a yuan quadratic equation on λ, we can get
is the eigenvalue of
, that is
, the eigenvalues of H are all positive and real numbers, the corresponding eigenfunction have the form
. For (3.13) and future reference, we observe that for all
is uniformly bounded and globally Lipschitz continuous.
, we get
. According to (H1), we can obtain
Theorem 3.1 If
is maximum Lipschitz constant of
, and if
is sufficiently large such that when
, the following inequality holds:
Then the operator H satisfies the spectral gap condition of (2.6).
, all the eigenvalues of H are real and positive, and
we can easily know that both sequences
The whole process of proof is divided into four steps.
Step 1. Since
is arranged in nondecreasing order. According to Lemma 2.1, given N such that
are consecutive, we separate the eigenvalue of H as
Step 2. We make decomposition of X
In order to make these two subspaces orthogonal and satisfy spectral inequality (2.6)
, we further decompose
. Next, we stipulate an eigenvalue scale product of X such that
are orthogonal, therefore we need to introduce two functions:
respectively are the conjugation of
holds, we can know
. Therefore, for all
, analogously, for all
, we can get
From above, we know that for all
holds. So, we define a scale product with Φ and ψ in X.
are respectively the projection:
In the inner product of X in (3.29),
are orthogonal. In fact, we need prove that
, we can get
the above results.
Step 3. Next, we estimate the Lipschitz constant
are globally Lipschitz continuous with
maximum Lipschitz constant
from (3.27), (3.28), for arbitrarily
, we have.
are the orthogonal projection.
Step 4. Now, we need prove the spectral gap condition (2.6) holds.
From the above mentioned
, we can get
such that for all
we can get
According to the previous hypothesis
, we can know
Then, combining (3.33), (3.34), (3.17) and (3.37), we obtain
The proof is completed.
Theorem 3.2.  Under the condition of Theorem 3.1, the initial boundary value problem (1.1)-(1.6) admits an inertial manifold μ in X of the form
are as in (3.20), (3.21) and
is a Lipschitz continuous function.
In this paper, we prove the existence of the inertial manifolds for a class of higher-order coupled Kirchhoff-type equations. In the process of research, we take advantage of Hadamard’s graph to get the equivalent form of the original equations and then use spectral gap condition. Based on some of the work above, we prove the existence of the inertial manifolds of the system. For this problem, we will study the exponential attractors, blow-up, random attractors and so on.
We express our heartful thanks to the anonymous reader for his/her careful reading of this paper. We hope that we can obtain valuable comments and advices. These contributions vastly improved the paper and making the paper better.