In this paper, we concerned the equation:

{ u t t − M ( ‖ ∇ u ‖ 2 + ‖ ∇ m v ‖ 2 ) Δ u − β Δ u t + g 1 ( u , v ) = f 1 ( x ) , v t t + M ( ‖ ∇ u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m v + β ( − Δ ) m v t + g 2 ( u , v ) = f 2 ( x ) , u ( x , 0 ) = u 0 ( x ) ,   u t ( x , 0 ) = u 1 ( x ) ,   x ∈ Ω , v ( x , 0 ) = v 0 ( x ) ,   v t ( x , 0 ) = v 1 ( x ) ,   x ∈ Ω , u | ∂ Ω = 0 ,   v | ∂ Ω = 0 ,   ∂ i v ∂ μ i | ∂ Ω = 0     ( i = 1 , 2 , ⋯ , m − 1 ) , (1.1)

where Ω is a bounded domain in Rⁿ with a smooth boundary ∂ Ω , β > 1 is a constant and f i ( x ) ( i = 1 , 2 ) is a given source term. Moreover, M ( ‖ ∇ u ‖ 2 + ‖ ∇ m v ‖ 2 ) is a scalar function. Then the assumptions on M and g i ( u , v ) will be specified later.

Nowadays, the study on the complexity of the space-time of high dimensional and infinite dimensional dynamical systems has gradually become the focus of nonlinear scientific research. In recent years, the inertial manifold has been found in the researches of the long time behavior of the solution and the attractor structure. The inertial manifold is a tool to describe the interaction between the low frequency components and the high frequency components [1] . When the flow has an inertial manifold, its high frequency description depends on the low frequency, and it contains attractors and exponentially attracts solution of the track, which realizes that the infinite dimensional dynamical system is reduced to a finite dimensional dynamical systems of the finite dimensional invariable Lipschitz manifold. Therefore, the inertial manifold is a powerful tool to study the long-time behavior of nonlinear dissipative systems and expose the real or seemingly chaotic structure of nonlinear dynamics.

In addition, the study of inertial manifold is of great significance. The central idea of the methods that people use to solve practical problems such as Galerkin method, Cellular automaton and Coupled map, are to discuss the infinite dimensional problem into a finite dimensional problem. So, the inertial manifold is of great significance to the development of nonlinear science.

In 1988, the concept of inertial manifold was first proposed in the study of infinite dimensional dynamical system by R. Temam, C. Foias and Sell G.R. [2] . They considered the equation as following:

u t + A u + B ( u , u ) + C ( u ) − f = 0 . (1.2)

where Au is a linear unbounded self-adjoint operator on H with domain D ( A ) dense in H.

In 2010, Guoguang Lin and Jingzhu Wu [3] studied the existence of the inertial manifold of Boussinesq equation:

{ u t t − α Δ u t − Δ u + u 2 k + 1 = F ( x , y ) , u ( x , y , 0 ) = u 0 ( x , y ) , u ( x , y , t ) = u ( x + π , y , t ) = u ( x , y + π , t ) = 0 , ( x , y ) ∈ Ω , (1.3)

where Ω = ( 0 , π ) 2 ∈ R 2 , t > 0 , α > 2 .

In 2016, Ling Chen, Wei Wang and Guoguang Lin [4] established the exponential attractors and inertial manifolds of the higher-order Kirchhoff-type equation:

u t t + ( − Δ ) m u t + ϕ ( ‖ ∇ m u ‖ 2 ) ( − Δ ) m u + g ( u ) = f ( x ) . (1.4)

There are many researches on inertial manifolds for nonlinear wave equations (see [5] [6] ). Concerning the inertial manifold, many difficulties are solved. So we take advantage of Hadamard’s graph transformation method in this paper.

The paper is arranged as follows. In Section 2, some assumptions, notations and lemmas are stated. In Section 3, the existence of the inertial manifold is established.

For convenience, we first introduce the following notations:

X = H 0 2 ( Ω ) × H 0 2 m ( Ω ) × H 0 1 ( Ω ) × H 0 m ( Ω ) ,   X 0 = H 0 1 ( Ω ) × H 0 m ( Ω ) ,

c i ( i = 1 , 2 , ⋯ ) denotes different positive constants, ( ⋅ , ⋅ ) and ‖   ⋅   ‖ are the inner product and norm of L 2 ( Ω ) , ‖   ⋅   ‖ − m is the norm of H − m ( Ω ) .

Next, we give some assumptions and definition needed in the proof of our results.

( A 1 )   g i ( u , v ) ∈ C 1 ( Ω ) ,   ( i = 1 , 2 ) .

( A 2 )   ε ≤ m 0 ≤ M ( s ) ≤ m 1 ≤ ( β − 1 ) μ k 4 .

Definition 2.1. [7] Let A : X → X be an operator and assume that F ∈ C b ( X , X ) satisfies the Lipschitz condition

‖ F ( U ) − F ( V ) ‖ X ≤ l F ‖ U − V ‖ X ,   U , V ∈ X . (2.1)

The operator A is called satisfy the spectral gap condition relative to F, if the point spectrum of the operator A can be divided into two parts σ 1 and σ 2 , of which σ 1 is finite, and such that, if

Λ 1 = sup { Re λ | λ ∈ σ 1 } ,   Λ 2 = inf { Re λ | λ ∈ σ 2 } , (2.2)

and

X i = s p a n { w j | λ j ∈ σ i } ,   i = 1 , 2 . (2.3)

Then

Λ 2 − Λ 1 > 4 l F , (2.4)

and the orthogonal decomposition X = X 1 ⊕ X 2 holds with continuous orthogonal projections P 1 : X → X 1 , P 2 : X → X 2 .

Lemma 2.1. [8] Let the eigenvalues μ j ± , j ≥ 1 be arranged in nondecreasing order. For all ∀ m ∈ N , there exists N ≥ m such that μ N − and μ N + 1 − are consecutive.

Equation (1.1) is equivalent to the following one order evolution equation:

U t + A U = F ( U ) , (3.1)

where U = ( u , v , p , q ) T ∈ X , p = u t , q = v t ,

A = ( 0 0 − I 0 0 0 0 − I − M ( s ) Δ 0 − β Δ 0 0 M ( s ) ( − Δ ) m 0 β ( − Δ ) m ) , F ( U ) = ( 0 0 f 1 ( x ) − g 1 ( u , v ) f 2 ( x ) − g 2 ( u , v ) ) (3.2)

D ( A ) = { ( u , v ) ∈ H 0 2 ( Ω ) × H 0 2 m ( Ω ) } × H 0 1 ( Ω ) × H 0 m ( Ω ) .

We consider the usual graph norm in X, as follows

( U , V ) X = ( M ( s ) ∇ u , ∇ u 1 ¯ ) + ( M ( s ) ∇ m v , ∇ m v 1 ¯ ) + ( p 1 ¯ , p ) + ( q 1 ¯ , q ) , (3.3)

where U = ( u , v , p , q ) T ,   V = ( u 1 , v 1 , p 1 , q 1 ) T ∈ X ,   u 1 ¯ ,   v 1 ¯ ,   p 1 ¯ ,   q 1 ¯ respectively represent the conjugation of u 1 , v 1 , p 1 , q 1 . Evidently, the operator A is monotone, for U ∈ D ( A ) , we obtain

( A U , U ) X = ( − M ( s ) ∇ p , ∇ u ¯ ) + ( − M ( s ) ∇ m q , ∇ m v ¯ )     + ( p ¯ , − M ( s ) Δ u − β Δ p ) + ( q ¯ , M ( s ) ( − Δ ) m v + β ( − Δ ) m q ) = ( − M ( s ) ∇ p , ∇ u ¯ ) + ( ∇ p ¯ , M ( s ) ∇ u ) + β ‖ ∇ p ‖ 2     + ( − M ( s ) ∇ m q , ∇ m v ¯ ) + ( ∇ m q ¯ , M ( s ) ∇ m v ) + β ‖ ∇ m q ‖ 2 = β ‖ ∇ p ‖ 2 + β ‖ ∇ m q ‖ 2 ≥ 0. (3.4)

So, ( A U , U ) X is a nonnegative and real number.

In order to determine the eigenvalues of A, we consider the eigenvalues equation:

A U = λ U ,   U = ( u , v , p , q ) T ∈ X . (3.5)

That is

{ − p = λ u ,       (3 .6) − q = λ v ,     (3 .7) − M ( s ) Δ u − β Δ p = λ p , (3 .8) M ( s ) ( − Δ ) m v + β ( − Δ ) m q = λ q . (3 .9)

Substitute (3.6), (3.7) into (3.8), (3.9), we obtain

{ λ 2 u − M ( s ) Δ u + β λ Δ u = 0 ,   (3 .9) λ 2 v + M ( s ) ( − Δ ) m v − β λ ( − Δ ) m v = 0. (3.10)

Replacing u , v with u k , v k , taking u , v inner product with the Equations (3.9), (3.10), and adding them together, we have

λ k k ( ‖ u k ‖ 2 + ‖ v k ‖ 2 ) + M ( s ) ( ‖ ∇ u k ‖ 2 + ‖ ∇ m v k ‖ 2 ) − λ k β ( ‖ ∇ u k ‖ 2 + ‖ ∇ m v k ‖ 2 ) = 0 . (3.11)

(3.11) is regard as a quadratic equation with one unknown about λ k , so we get

λ k ± = β μ k ± β 2 μ k 2 − 4 M ( s ) μ k 2 , (3.12)

for ∀ k ≥ 1 , we have

‖ u k ‖ 2 + ‖ v k ‖ 2 = 1 , ‖ ∇ u k ‖ 2 + ‖ ∇ m v k ‖ 2 = μ k , ‖ ∇ − 1 u k ‖ 2 + ‖ ∇ − m v k ‖ 2 = 1 μ k . (3.13) and μ k is non-derogatory. If β ≥ 4 M ( s ) μ k + 1 , because of β > 1 , then β 2 ≥ 4 M ( s ) μ k , we can get the eigenvalues of A are all positive and real numbers.

The corresponding eigenfunction is as follows

U k ± = ( u k , v k , − λ k ± u k , − λ k ± v k ) . (3.14)

Lemma 3.1. g i : X 0 → X 0 , ( i = 1 , 2 ) is uniformly bounded and globally Lipschitz continuous.

Proof. ∀ ( u , v ) , ( u 1 , v 1 ) ∈ X 0 , by ( A 1 ) , we have

‖ g i ( u , v ) − g i ( u 1 , v 1 ) ‖ X 0 ≤ ‖ g i u ( ξ , v ) ( u − u 1 ) ‖ H 0 1 ( Ω ) + ‖ g i v ( u , η ) ( v − v 1 ) ‖ H 0 m ( Ω ) ≤ c i 1 ‖ u − u 1 ‖ H 0 1 ( Ω ) + c i 2 ‖ v − v 1 ‖ H 0 m ( Ω ) ≤ c i 3 ( ‖ u − u 1 ‖ H 0 1 ( Ω ) + ‖ v − v 1 ‖ H 0 m ( Ω ) ) , (3.15)

where

c i 3 = max { c i 1 , c i 2 } ,   ξ = u + ( 1 − θ 1 ) u 1 ,   η = v + ( 1 − θ 2 ) v 1 .

Let l i = c i 3 , then l i is Lipschitz coefficient of g i ( u , v ) .

Theorem 3.1. l i is Lipschitz constant of g i ( u , v ) , when β ≥ 4 M ( s ) μ k + 1 , set N 1 ∈ N , such that N ≥ N 1 , we obtain

( μ N + 1 − μ N ) ( 1 2 − 1 2 ( β − 1 ) μ 1 − 4 m 1 ) ≥ 4 l ( β − 1 ) μ 1 − 4 m 1 + 1 , (3.16)

where l = max { l 1 , l 2 } . By ( A 2 ) and Lemma 3.1, the operator A satisfies the spectral gap condition of (2.4).

Proof. when β ≥ 4 M ( s ) μ k + 1 , the eigenvalues of A are all positive and real

numbers, meanwhile { λ k − } k ≥ 1 and { λ k + } k ≥ 1 are increasing order.

Next, we divided the whole process of proof into four steps.

Step 1 By Lemma 2.1, since { λ k ± } is nondecreasing order, so there exists N, such that λ N − and λ N + 1 − are continuous. Then the eigenvalues of A are separate as

σ 1 = { λ j − , λ k + | max { λ j − , λ k + } ≤ λ N − } , σ 2 = { λ j − , λ k ± | λ j − ≤ λ N − ≤ min { λ j + , λ k ± } } . (3.17)

Step 2 The corresponding X is decomposed into

X 1 = s p a n { U j − , U k + | λ j − , λ k + ∈ σ 1 } , X 2 = s p a n { U j − , U k ± | λ j − , λ k ± ∈ σ 2 } . (3.18)

We aim at madding two orthogonal subspaces of X and verifying the spectral gap condition (2.4) is true when Λ 1 = λ N − , Λ 2 = λ N + 1 − . Therefore, we further decompose X 2 = X c ⊕ X R , where

X c = s p a n { U j − | λ j − ≤ λ N − < λ j + } , X R = s p a n { U k ± | λ N − < λ k ± } . (3.19)

Set X N = X 1 ⊕ X c , in order to verify the X 1 and X 2 are orthogonal, we need to introduce two functions Φ : X N → R , Ψ : X R → R .

Φ ( U , V ) = β ( ∇ u , ∇ u 1 ¯ ) − 4 M ( s ) ( u , u 1 ¯ ) + ( ∇ − 1 p 1 ¯ , ∇ u ) + ( ∇ − 1 p ¯ , ∇ u 1 )     +4 ( ∇ − 1 p , ∇ − 1 p 1 ¯ ) + β ( ∇ m v , ∇ m v 1 ¯ ) − 4 M ( s ) ( v , v 1 ¯ )     + ( ∇ − m q 1 ¯ , ∇ m v ) + ( ∇ − m q ¯ , ∇ m v 1 ) + 4 ( ∇ − m q , ∇ − m q 1 ¯ ) , (3.20)

Ψ ( U , V ) = 2 β ( ∇ u , ∇ u 1 ¯ ) − β ( ∇ − 1 p 1 ¯ , ∇ u ) − β ( ∇ − 1 p ¯ , ∇ u 1 )     + β ( ∇ − 1 p , ∇ − 1 p 1 ¯ ) + 2 β ( ∇ m v , ∇ m v 1 ¯ ) − β ( ∇ − m q 1 ¯ , ∇ m v )     − β ( ∇ − m q ¯ , ∇ m v 1 ) + β ( ∇ − m q , ∇ − m q 1 ¯ ) , (3.21)

where U , V ∈ X are defined before.

Let U = ( u , v , p , q ) ∈ X N , by ( A 2 ) , then

Φ ( U , U ) = β ( ∇ u , ∇ u ¯ ) − 4 M ( s ) ( u , u ¯ ) + ( ∇ − 1 p ¯ , ∇ u ) + ( ∇ − 1 p ¯ , ∇ u )     +4 ( ∇ − 1 p , ∇ − 1 p ¯ ) + β ( ∇ m v , ∇ m v ¯ ) − 4 M ( s ) ( v , v ¯ )     + ( ∇ − m q ¯ , ∇ m v ) + ( ∇ − m q ¯ , ∇ m v ) + 4 ( ∇ − m q , ∇ − m q ¯ ) ≥ ( β − 1 ) ( ‖ ∇ u ‖ 2 + ‖ ∇ m v ‖ 2 ) − 4 M ( s ) ( ‖ u ‖ 2 + ‖ v ‖ 2 ) ≥ [ ( β − 1 ) μ 1 − 4 m 1 ] ( ‖ u ‖ 2 + ‖ v ‖ 2 ) . (3.22)

Since for ∀ k , m 1 ≤ ( β − 1 ) μ k 4 , therefore Φ ( U , U ) ≥ 0 , for ∀ U ∈ X N , then Φ is positive definite.

Similarly, for U ∈ X R , we have

Ψ ( U , U ) = 2 β ( ∇ u , ∇ u ¯ ) − β ( ∇ − 1 p ¯ , ∇ u ) − β ( ∇ − 1 p ¯ , ∇ u )     + β ( ∇ − 1 p , ∇ − 1 p ¯ ) + 2 β ( ∇ m v , ∇ m v ¯ ) − β ( ∇ − m q ¯ , ∇ m v )     − β ( ∇ − m q ¯ , ∇ m v ) + β ( ∇ − m q , ∇ − m q ¯ ) ≥ β ( ‖ ∇ u ‖ 2 + ‖ ∇ m v ‖ 2 ) ≥ β μ 1 ( ‖ u ‖ 2 + ‖ v ‖ 2 ) . (3.23)

So, for ∀ U ∈ X R , Ψ ( U , U ) ≥ 0 , the Ψ is also positive definite.

Next, we need to define a scale product in X

〈 〈 U , V 〉 〉 X = Φ ( P N U , P N V ) + Ψ ( P R U , P R V ) , (3.24)

where P N and P R are projection: X → X N , X → X R respectively, for convenience, we rewrite (3.24) as follows

〈 〈 U , V 〉 〉 X = Φ ( U , V ) + Ψ ( U , V ) . (3.25)

We will proof that two subspaces X₁ and X₂ in (3.18) are orthogonal. In fact, we only need to show X_N and X_C are orthogonal, that is

〈 〈 U j − , U j + 〉 〉 X = 0   ( ∀ U j − ∈ X N , U j + ∈ X c ) . (3.26)

By (3.20), (3.25), we have

〈 〈 U j − , U j + 〉 〉 X = Φ ( U j − , U j + ) = β ( ∇ u j , ∇ u j ¯ ) − 4 M ( s ) ( u j , u j ¯ ) − λ j + ( ∇ − 1 u j ¯ , ∇ u j ) − λ j − ( ∇ − 1 u j ¯ , ∇ u j )     + 4 λ j − λ j + ( ∇ − 1 u j , ∇ − 1 u j ¯ ) + β ( ∇ m v j , ∇ m v j ¯ ) − 4 M ( s ) ( v j , v j ¯ )     − λ j + ( ∇ − m v j ¯ , ∇ m v j ) − λ j − ( ∇ − m v j ¯ , ∇ m v j ) + 4 λ j − λ j + ( ∇ − m v j , ∇ − m v j ¯ ) = β ( ‖ ∇ u j ‖ 2 + ‖ ∇ m v j ‖ 2 ) − M ( s ) ( ‖ u j ‖ 2 + ‖ v j ‖ 2 )     − ( λ j + + λ j − ) ( ‖ u j ‖ 2 + ‖ v j ‖ 2 ) + 4 λ j − λ j + ( ‖ u j ‖ − 1 2 + ‖ v j ‖ − m 2 ) . (3.27)

By (3.12), we can get λ j + + λ j − = β μ j , λ j − λ j + = M ( s ) μ j , therefore

〈 〈 U j − , U j + 〉 〉 X = Φ ( U j − , U j + ) = 0 . (3.28)

Step 3 Further, we estimate the Lipschitz constant l F of F ( U ) (3.2). According to Lemma 3.1, g i : X 3 → X 3 is Lipschitz continuous with Lipschitz constant l i . Let P i : X → X i ( i = 1 , 2 ) is orthogonal projection. From (3.22), (3.23) and (3.24), we have

‖ U ‖ X 2 = Φ ( P 1 U , P 1 U ) + Ψ ( P 2 U , P 2 U ) ≥ [ ( β − 1 ) μ 1 − 4 m 1 ] ( ‖ P 1 u ‖ 2 + ‖ P 1 v ‖ 2 ) + β μ 1 ( ‖ P 2 u ‖ 2 + ‖ P 2 v ‖ 2 ) ≥ [ ( β − 1 ) μ 1 − 4 m 1 ] ( ‖ P 1 u ‖ 2 + ‖ P 2 u ‖ 2 + ‖ P 1 v ‖ 2 + ‖ P 2 v ‖ 2 ) ≥ [ ( β − 1 ) μ 1 − 4 m 1 ] ( ‖ u ‖ 2 + ‖ v ‖ 2 ) . (3.29)

Given U = ( u , v , p , q ) T , V = ( u 1 , v 1 , p 1 , q 1 ) T ∈ X , we have

‖ F ( U ) − F ( V ) ‖ X = ‖ g 1 ( u , v ) − g 1 ( u 1 , v 1 ) ‖ X 3 + ‖ g 2 ( u , v ) − g 2 ( u 1 , v 1 ) ‖ X 3 ≤ l 1 ( ‖ u − u 1 ‖ + ‖ v − v 1 ‖ ) + l 2 ( ‖ u − u 1 ‖ + ‖ v − v 1 ‖ ) ≤ l ( ‖ u − u 1 ‖ + ‖ v − v 1 ‖ ) ≤ l ( β − 1 ) − 4 m 1 ‖ U − V ‖ X , (3.30)

where l = max { l 1 , l 2 } .

So, we obtain

l F ≤ l ( β − 1 ) − 4 m 1 . (3.31)

Step 4 Now, we will show the spectral gap condition (2.4) holds.

Since Λ 1 = λ N − , Λ 2 = λ N + 1 − , then

Λ 2 − Λ 1 = λ N + 1 − − λ N − = 1 2 ( μ N + 1 − μ N ) + 1 2 Q ( N ) − Q ( N + 1 ) , (3.32)

where Q ( N ) = β 2 μ N 2 − 4 M ( s ) μ N .

Let N 1 > 0 , for ∀ N ≥ N 1 , then Q 1 ( N ) = 1 − β 2 ( β − 1 ) μ 1 − 4 m 1 − 4 m 1 μ N [ ( β − 1 ) μ 1 − 4 m 1 ] , we can obtain

Q ( N ) − Q ( N + 1 ) + ( β − 1 ) − 4 m 1 ( μ N + 1 − μ N ) = ( β − 1 ) μ 1 − 4 m 1 ( μ N + 1 Q 1 ( N + 1 ) − μ N Q 1 ( N ) ) (3.33)

From ( A 2 ) , we can easily obtain

lim N → ∞ ( Q ( N ) − Q ( N + 1 ) + ( β − 1 ) μ 1 − 4 m 1 ( μ N + 1 − μ N ) ) = 0 . (3.34)

Then, according to (3.16), (3.31), (3.32) and (3.34), we have

Λ 2 − Λ 1 > ( μ N + 1 − μ N ) ( 1 2 − 1 2 ( β − 1 ) μ 1 − 4 m 1 ) − 1 ≥ 4 l ( β − 1 ) μ 1 − 4 m 1 ≥ 4 l F . (3.35)

Therefore, Theorem 3.1 is true.

Theorem 3.2. Under the condition of Theorem 3.1, the problem (1.1)-(1.5) exist an inertial manifold μ in X,

μ = g r a p h ( m ) : = { ζ + m ( ζ ) : ζ ∈ X 1 } , (3.36)

where X 1 , X 2 defined in (3.18) and m : X 1 → X 2 is a Lipschitz continuous function.

The authors would like to thank for the anonymous referees for their valuable comments and suggestions sincerely. These contributions increase the value of the paper.

Lin, G.G. and Xia, X.S. (2018) The Inertial Manifold for Class Kirchhoff-Type Equations with Strongly Damped Terms and Source Terms. Applied Mathematics, 9, 730-737. https://doi.org/10.4236/am.2018.96050