Inertial Manifolds for Generalized Higher-Order Kirchhoff Type Equations

The existence of inertial manifolds for higher-order Kirchhoff type equations with strong damping terms is studied. The Hadamard graph norm conversion method is used for obtaining the existence of inertial manifolds for this kind of equations under certain spectral intervals.


Introduction
In the study of the long-term dynamic behavior of infinite dimensional dynamical systems, the inertial manifold occupies an important position. It is a finite dimensional invariant Lipschitz manifold and attracts all solution orbitals with exponential rate in the phase space of the system [1][2][3]. It plays an important role in both finite dimensional dynamical systems and infinite dimensional dynamical systems.Because it occupies an important position, many scholars have studied the existence and attraction of inertial manifolds, the finite-dimensional properties, and the related problems of approximate inertial manifolds and delay. Guoguang Lin and Jingzhu Wu [4] studied the existence of inertial manifolds of the low order Bousinesq equation with strongly damped term at that time, and the equation is u(x, y, 0) = u 0 (x, y), (x, y) ∈ Ω, u(x, y, t) = u(x + π, y, t) = u(x, y + π, t) = 0, (x, y) ∈ Ω.
Zhicheng Zhang and Guoguang Lin [5] study the following fourth order strongly damped time-delay wave equations: The inertial manifolds of the above equations under the assumption of delay term from distribution are studied. In this paper, based on this, will rise again, order space optimization, joined the high order structural damping, under certain assumptions prove Kirchhoff type generalized high order equation of inertial manifolds exist. More on the Kirchhoff equation of inertial manifolds, see reference [7][8][9][10][11][12].
This paper study the initial boundary value problems of the following Kirchhoff type equations: Where r ≥ 0, m ≥ 1, β > 0. Ω is the bounded region with smooth boundary ∂Ω in R n . f (x) is the external force term. v i is the external normal vector,∆ 2m u is the structural damping term,β(−∆) m u t is the structure damping term. (1 + Ω |D m u| p dx) r (−∆) m u is the rigid term. And the assumptions about the rigid term will be given late.

Prepare
For the convenience, this paper defines Spaces and symbols as follows : × H k 0 (Ω)(k = 0, 1, 2, · · · , 2m), C 0 is constant. Respectively,(·, ·) and · represent the inner product and norm of H. That is (u Vol. 12, No. 5;2020 u 2 .And then let's define the norm and the inner product of the spaces V 1 and V 2 : g( D m u p p ) = (1 + Ω |D m u| p dx) r meets the following conditions: Definition 2.1[2] One says that inertial manifolds µ are finite-dimensional manifolds that need to satisfy the following these properties: (i) µ is a lipschitz manifold and finite dimension; The µ exponential attracts all the solution orbitals.
Definition 2.2[2]Assume A 1 : X → X is an operator and F ∈ C b (X, X) satisfy the following inequalities Suppose the point spectrum of operator A 1 can be divided into two parts σ 1 and σ 2 , and which σ 1 is finite, And So We have orthogonal decomposition and continuous mapping Lemma 2.1[6]Suppose that the eigenvalues µ ± j ( j ≥ 1) are nonsubtractive, and for all m ∈ N, when N ≥ m, µ − N and µ − N+1 are consecutive adjacent values.

Inertial Manifolds
Equation (1) is equivalent to the following first-order evolution equation where U = (u, v), v = u t , The graph norm defined in X by the dot product where U = (u, v), V = (y, z) ∈ X, u, y ∈ H 2m+k (Ω); v, z ∈ H 2m+k (Ω). y, z respectively represent conjugate of y, z. Obviously, the operator defined in equation (11) So ( AU, U) X is a non-negative real number.
To determine the eigenvalues of A,consider the following eigenvalue equation That is By substituting the first equation of fomular (17) into the second equation of equation (17), we obtained Used for the inner product of the first expression in equation (18), we get Equation (19) is regarded as a quadratic equation with one unknown about λ, so there is Where δ k is the eigenvalue of (−∆) m in H 2m (Ω). If β 2 ≥ 4 , then all the eigenvalues of A are positive real numbers, and the corresponding eigenvector has the form of U ± k = (u k , −λ ± k u k ) .As for formula (13), in order to facilitate the use of the following, the following marks will be made. For all k ≥ 1,we get Lemma3.1 Remarking g(u) = (1 + Ω |D m u| p dx) r (−∆) m u, g : H 2m+k 0 (Ω) → H 2m 0 (Ω) is uniformly bounded, and globally lipschitz continuous.
Theorem 3.1 When 0 < β ≤ 2, l is the lipschitz coefficient of g( D m u p p ), let N 1 ∈ N make that N ≥ N 1 , we have Then the operator A satisfies the (7) spectral interval condition.
Proof. According to equations (12) and (14), writing U = (u, v), V = ( u, v) ∈ X, then That is l F ≤ l. According to equation (19), the necessary and sufficient condition for λ ± k to be λ ± k real number that is β > 2. By assuming that 0 < β ≤ 2, A have at most λ ± k finite number of 2N 0 as eigenroots, when N 0 = 0, 0 < β ≤ 2, then Λ 0 = max{λ ± k |k ≤ N 0 . When k ≥ N 0 + 1, the eigenvalue is complex, and the real part is taken So there is N 1 ≥ N 0 + 1 make Reλ ± k > Λ 0 , k ≥ N 1 . Let make (22) be true. Decompose the dot spectrum of A Let's set the corresponding subspace Inexistence k make δ − k ∈ σ 1 and δ + k ∈ σ 2 , which means it can't exist U − k ∈ X 1 and U + k ∈ X 2 . Therefore, X 1 and X 2 are the normal subspace of X. According to (5) and (25), we get Therefore, it can be known from (23) that A satisfies the spectral interval condition.
Theorem 3.2 When β ≥ 2, l is the Lipschtiz coefficient of g( D m u p p ) let N 1 ∈ N be sufficiently large, so that N ≥ N 1 and Then operator A satisfies the spectral interval condition of (7).
Proof. When β > 2, all eigenvalues of A are positive real numbers, and we know that the sequences {λ − k } k≥1 and {λ + k } k≥1 are increasing.The following are four steps to prove Theorem 3.2.
Step 1: Because λ ± k is non-subtractive, according to Lemma 2.1, N is given so that λ − N and λ − N+1 are adjacent values, and the eigenvalue of A is decomposed to Step 2: Corresponding X can be decomposed into The goal is to make these two subspaces orthogonal and satisfy the interspectral expression (7).
And assuming that X N = X C ⊕X 1 . Next, specify the dot product of the eigenvalues over X, so that X 1 and X 2 are orthogonal, so we need to introduce two functions Φ : X N → R and Ψ : X R → R.