A Family of Inertial Manifolds of Coupled Kirchhoff Equations

In this paper, we study the long-time behavior of the solution of the initial boundary value problem of the coupled Kirchhoff equations. Based on the relevant assumptions, the equivalent norm on k E is obtained by using the Hadamard graph transformation method, and the Lipschitz constant F l of F is further estimated. Finally, a family of inertial manifolds satisfying the spectral interval condition is obtained.


Introduction
This paper mainly studies the initial boundary value problem of the coupled Kirchhoff equations: , , ,0 , ,0 , , ,0 , ,0 0, 0, 0,1, 2, , 2 1 , . As we all know, an inertial manifold is a Lipschitz manifold that contains a global attractor and attracts all solution orbits at an exponential rate, and it is finite-dimensional and positively invariant. The inertial manifold is of great significance to study the long-term behavior of infinite dimensional dynamical systems. Because it transforms infinite dimensional problems into finite-dimensional problems, and an inertial manifold is of great significance to the development of nonlinear science.
In 1989, Constantin, Foias, Nicolaenko, et al. [1] tried to refine the spectral separation conditions by using the concept of spectral barrier in Hilbert space.
, , , , f x f x are the external force terms, are strong dissipative terms. Using the Hadamard graph transformation method, they obtain the existence of the inertial manifold while such equations satisfy the spectrum interval condition.
Guoguang Lin and Lujiao Yang in [4] first studied the family of inertial manifolds and exponential attractors for the Kirchhoff equations.
is real function. After making appropriate assumptions, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation. Then according to Hadamard's graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.
Because an inertial manifold plays a very important role in describing the long-time behavior of solutions, it is of great significance to the development of nonlinear science. The relevant research theoretical results are shown in references [5]- [19].
On the basis of previous studies, this paper further improves the order of the strong dissipative term and the rigid term mentioned by Guoguang Lin and Lingjuan Hu [3], where the coefficient of the rigid term is extended from , , , are new nonlinear terms. When constructing the equivalent norm in k E space, through reasonable assumptions and combined with the Lipschitz property of the nonlinear term, the family of inertial manifolds satisfying the spectral interval condition is obtained.

Preliminaries
The following symbols and assumptions are introduced for the convenience of statement: is a solution semigroup on Banach space is said to be a family of inertial manifolds, if they satisfy the following three properties: µ attracts exponentially all orbits of solution , that is, for any Definition 2.2 [5] Assuming the operator : have countable positive real part eigenvalues and satisfies the Lipschitz condition: If the point spectrum of the operator A can be divided into the following two and satisfy

The Family of Inertial Manifolds
From the above preparation knowledge, Equations (1)-(5) are equivalent to In order to determine the eigenvalue of operator A, we must first consider the graph norm generated by the inner product in k E Indeed, for To further determine the eigenvalues of A, we consider the following charac- Take ( ) ( ) The above Equation (13) The above Equation (14)  where k µ is the eigenvalue of ( ) ( ) , taking For the convenience of the following description, for any positive integer k, Lemma 3.1
Proof. for arbitrary ( ) ( ) , , , , , where l is the Lipschitz constant of ( )( ) The following is divided into four steps to prove this theorem. Step The corresponding k E can be decomposed into { } 2 2 , , In order to prove the spectral interval condition, we will find out the orthogonality of subspaces Note that Similarly,

( )
, , , The main calculation process is as follows Through Equation (14) = is uniformly bounded and globally Lipschitz continuous, and Step 4 Prove Equation (6)  l of F can be further estimated. Finally, Formula (26) holds and then operator A satisfies the spectral interval condition. Next, we will further obtain that the initial boundary value problems (1)-(5) have a family of inertial manifolds. is Lipschitz continuous function.