A Family of the Inertial Manifolds for a Class of Generalized Kirchhoff-Type Coupled Equations

The paper considers the long-time behavior for a class of generalized high-order Kirchhoff-type coupled equations, under the corresponding hypothetical conditions, according to the Hadamard graph transformation method, obtain the equivalent norm in space , and we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectral interval condition.


Introduction
This paper investigates the following primal value problems of a system of generalized Kirchhoff-type coupled equations: , , ,0 , ,0 , , ,0 , ,0 , , where Ω is a bounded region with a smooth boundary in n R , ∂Ω represents the boundary of Ω , In order to overcome the research difficulties, G. Foias, G. R. Sell and R. Temam [1] proposed the concept of inertial manifolds, which greatly promoted the study of infinite-dimensional dynamical systems. Where the inertial manifold is a positive, finite-dimensional Lipschitz manifold, and the existence of an inertial manifold depends on the establishment of a spectral interval condition. Therefore, the research on a family of inertial manifolds is of great significance from both theoretical and practical aspects, and the relevant theoretical achievements can be referred to [2]- [9].
Guoguang Lin, Lingjuan Hu [10] studied a system of coupled wave equations of higher-order Kirchhoff type with strong damping terms where Ω is a bounded region with a smooth boundary in n R , ∂Ω represents the boundary of Ω , , f x f x is an external force interference term, and ( ) β ≥ is a strong dissipation terms. Using the Hadamard graph transformation method, the Lipschitz constant F l of F is further estimated, and the inertial manifolds that satisfies the spectral interval condition is obtained.
Lin Guoguang, Liu Xiaomei [11] studied a family of inertial manifolds for a class of generalized higher-order Kirchhoff equations with strong dissipation

Preliminaries
For narrative convenience, we introduce the following symbols and assumptions: inner product and norm are ( ) respectively. Apparently there are The assumption is as follows:

A Family of Inertial Manifolds
Definition 1 [12] lets 3) k µ attracts the solution orbit exponentially, i.e. for any k u E ∈ , the exis- Then k µ is called k E is a family of inertial manifolds.
In order to describe the spectral interval condition, first consider that the The point spectrum of the operator A can be divided into two parts 1 σ and 2 σ , and 1 σ is finite, are satisfied.
Lemma 2 [12] lets the sequence of eigenvalues { } 1 In order to verify that the operator satisfies the spectral interval condition, so as to draw the conclusion that there is a family of inertial manifolds in questions (1)-(5), the following definitions and assumptions can be made first.
Based on the above relevant conditions, consider the first-order development equation equivalent to Equations (1)-(5), as follows: In order to determine the eigenvalue of matrix operator A′ , first consider graph module , , , and , , , u z v q ′ ′ ′ ′ represent the conjugation of , , , u z v q ′ ′ ′ ′ respectively. In addition, operator A′ is monotonic, and for ( ) In order to further determine the eigenvalue of the matrix operator A′ , the following characteristic equation can be considered, That is  (1) and (2) Equation (9)   The following four steps are taken to prove theorem 1: Step Construct two functions : , : Similarly, for any Redefine the inner product of k E : (11) where N P and R P are projections of k Here, Equation (11) is written as Because of Equation (9), there are ( ) , .
Step  Then the spectral interval condition (6) holds.
Step 4: according to the above paired eigenvalues, there are And because of Theorem 2 [12] Through theorem1, operator A′ satisfies the spectral interval condition, and problems (1)-(5) have a family of inertial manifolds k