^{1}

^{*}

^{1}

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, 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.

In the study of dynamic behavior for a long time in infinite dimensional dynamical system, the exponential attractors and inertial manifolds play a very important role. In 1994, Foias [

Guigui Xu, Libo Wang and Guoguang Lin [

( u t t − α Δ u + β Δ 2 u − γ Δ u t + g ( u ) = f ( x , t ) , ( x , t ) ∈ Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u | ∂ Ω = 0 , Δ u | ∂ Ω = 0 , ( x , t ) ∈ ∂ Ω × R + .

The assumption of g ( u ) satisfies the following conditions:

(H_{1}) l i m | s | → ∞ inf G ( s ) s 2 ≥ 0, s ∈ R , G ( s ) = ∫ 0 s g ( r ) d r ;

(H_{2}) There is a positive constant C 1 , such that l i m | s | → ∞ inf s g ( s ) − C 1 G ( s ) s 2 ≥ 0, s ∈ R .

Under these reasonable assumptions, according to Hadamard’s graph transformation method, the existence of the inertial manifolds for the equation is obtained.

Zhijian Yang and Zhiming Liu [

u t t − σ ( ‖ ∇ u ‖ 2 ) Δ u t − ϕ ( ‖ ∇ u ‖ 2 ) Δ u + f ( u ) = h ( x ) .

The main result was that the nonlinearity f ( u ) is of supercritical growth and they established an exponential attractor in natural energy space by using a new method based on the weak quasi-stability estimates.

Ruijin Lou, Penghui Lv, Guoguang Lin [

u t t + M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m u t + g ( u ) = f ( x ) ,

u ( x , t ) = 0 , ∂ i u ∂ v i = 0 , i = 1 , 2 , ⋯ , 2 m − 1 , x ∈ ∂ Ω ,

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , t > 0.

where Ω is finite region of R n , ∂ Ω is smooth boundary, u 0 ( x ) and u 1 ( x ) is initial value, ( − Δ ) m u t is strongly damped term, ϕ is stress term, g ( u ) is nonlinear source term.

On the basis of reference [

In this paper, we study the existence of exponential attractors and a family of the inertial manifolds for a class of generalized Kirchhoff-type equation with damping term:

u t t + M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m u t + g ( u ) = f ( x ) , (1.1)

u ( x , t ) = 0 , ∂ i u ∂ v i = 0 , i = 1 , 2 , ⋯ , 2 m − 1 , x ∈ ∂ Ω , (1.2)

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , t > 0. (1.3)

where m > 1 , p ≥ 2 , Ω ⊂ R n ( n ≥ 1 ) is a bounded domain with a smooth boundary ∂ Ω , M ( s ) ∈ C 2 ( [ 0, + ∞ ) ; R + ) is a real function, β ( − Δ ) 2 m u t ( β > 0 ) denotes strong damping term, g ( u ) is nonlinear source term, f ( x ) denotes the external force term. The assumption of M ( s ) and g ( u ) as follow:

(A1) g ( u ) ∈ C ∞ ( R )

(A2) M ( s ) ∈ C 2 ( [ 0 , + ∞ ) , R + ) , 1 ≤ μ 0 < M ( s ) < μ 1 , μ = { μ 0 , d d t ‖ ∇ 2 m u ‖ 2 ≥ 0 , μ 1 , d d t ‖ ∇ 2 m u ‖ 2 < 0.

where μ , μ 0 , μ 1 are constant, λ 1 is the first eigenvalue of − Δ with homogeneous Dirichlet boundary conditions on Ω .

For convenience, define the following spaces and notations H = L 2 ( Ω ) , H 0 2 m ( Ω ) = H 2 m ( Ω ) ∩ H 0 1 ( Ω ) , H 0 4 m ( Ω ) = H 4 m ( Ω ) ∩ H 0 1 ( Ω ) , H 0 2 m + k ( Ω ) = H 2 m + k ( Ω ) ∩ H 0 1 ( Ω ) , E 0 = H 2 m ( Ω ) × L 2 ( Ω ) , E k = H 0 2 m + k ( Ω ) × H 0 k ( Ω ) , ( k = 1 , 2 , ⋯ , 2 m ), f ( x ) ∈ L 2 ( Ω ) . ( ⋅ , ⋅ ) and ‖ ⋅ ‖ represent the inner product and norms of H respectively, i.e.

( u , v ) = ∫ Ω u ( x ) v ( x ) d x , ( u , u ) = ‖ u ‖ 2 , ‖ ⋅ ‖ = ‖ ⋅ ‖ L 2 ( Ω ) , ‖ ⋅ ‖ P = ‖ ⋅ ‖ L P ( Ω ) , ‖ ⋅ ‖ ∞ = ‖ ⋅ ‖ L ∞ ( Ω ) .

For brevity, define the inner product and norms as follow: ∀ U i = ( u i , v i ) ∈ E 0 , i = 1 , 2 ,

( U 1 , U 2 ) = ( ∇ 2 m u 1 , ∇ 2 m u 2 ) + ( v 1 , v 2 ) , (2.1)

‖ U ‖ E 0 2 = ( U , U ) E 0 = ‖ ∇ 2 m u ‖ 2 + ‖ v ‖ 2 . (2.2)

Let U = ( u , v ) ∈ E 0 , v = u t + ε u , 0 < ε ≤ min { 4 − β , 1 − α β λ 1 2 m , α 1 + λ 1 − 2 m } , we can get the Equation (1.1) is equivalent to the following evolution equation

U t + G ( U ) = F ( U ) . (2.3)

where

G = ( ε − I ε 2 − β ε ( − Δ ) 2 m + M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m β ( − Δ ) 2 m − ε ) ,

F ( U ) = ( 0 f ( x ) − g ( u ) ) .

Then, we will use the following notations. Let E 0 , E k are two Hilbert spaces, we have E k ↪ E 0 with dense and continuous injection, and E k ↪ E 0 is compact. Let S ( t ) is a map from E 0 ( E k ) into E 0 ( E k ) .

In the following definitions, k = 1 , 2 , ⋯ , 2 m .

Definition 2.1. [

Definition 2.2. [

Definition 2.3. [

‖ S ( t ) u − S ( t ) v ‖ E 0 ≤ l ( t ) ‖ u − v ‖ E 0 , ∀ ( u , v ) ∈ B k . (2.4)

Then the semigroup S ( t ) is Lipschitz continuous in B k .

Definition 2.4. [

‖ S ( t ∗ ) u − S ( t ∗ ) v ‖ E 0 ≤ δ ‖ u − v ‖ E 0 , (2.5)

or

‖ Q N ( S ( t ∗ ) u − S ( t ∗ ) v ) ‖ E 0 ≤ ‖ P N ( S ( t ∗ ) u − S ( t ∗ ) v ) ‖ E 0 (2.6)

Then S ( t ) is said to satisfy the discrete squeezing property, where Q N = I − P N .

Theorem 2.1. [

Proposition 2.1. [

the positive invariant set of S ( t ) in E 0 , and B k attracts all bounded subsets of E k , where B k is a closed bounded absorbing set for S ( t ) in E k .

Theorem 2.2. [

‖ ( u , v ) ‖ E 0 2 = ‖ ∇ 2 m u ‖ 2 + ‖ v ‖ 2 ≤ c ( r 0 ) , ‖ ( u , v ) ‖ E k 2 = ‖ ∇ 2 m + k u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ≤ c ( r 1 ) .

We denote the solution in Theorem 2.1 by S ( t ) ( u 0 , v 0 ) = ( u ( t ) , v ( t ) ) . Then S ( t ) composes a continuous semigroup in E 0 . According to Theorem 2.1, we have the ball

D 0 = { ( u , v ) ∈ E 0 : ‖ ( u , v ) ‖ E 0 2 = ‖ ∇ 2 m u ‖ 2 + ‖ v ‖ 2 ≤ c ( r 0 ) } , (2.7)

D k = { ( u , v ) ∈ E k : ‖ ( u , v ) ‖ E k 2 = ‖ ∇ 2 m + k u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ≤ c ( r 1 ) } . (2.8)

are absorbing sets of S ( t ) in E 0 and E k respectively. From Proposition 2.1

B k = ∪ 0 ≤ t ≤ t 0 ( D k ) S ( t ) D k ¯ . (2.9)

is a positive invariant compact set of S ( t ) in E 0 , and absorbs all of the bounded subsets D k in E k . According to reference [

A k = ∩ s ≥ 0 ∪ t ≥ s S ( t ) D k ¯ , where the bar means the closure in E 0 , and A k is bounded in E k .

Lemma 2.1. For any U = ( u , v ) ∈ E 0 ,

( G ( U ) , U ) E 0 ≥ a 1 ‖ U ‖ E 0 2 + a 2 ‖ ∇ 2 m v ‖ 2 . (2.10)

Proof. By (2.1) and (2.2), we have

( G ( U ) , U ) E 0 = ε ‖ ∇ 2 m u ‖ 2 − ( ∇ 2 m v , ∇ 2 m u ) + ( α − ε ) ‖ v ‖ 2 + ε 2 ( u , v ) − α ‖ v ‖ 2 − β ε ( ∇ 2 m u , ∇ 2 m v ) + β ‖ ∇ m v ‖ 2 + M ( ‖ ∇ m u ‖ p p ) ( ∇ 2 m u , ∇ 2 m v ) . (2.11)

By using Holder’s inequality, Young’s inequality and Poincare’s inequality and the condition (A2), we have,

ε 2 ( u , v ) = − ε 2 λ 1 − m ‖ ∇ 2 m u ‖ ‖ v ‖ ≥ − ε 2 4 ‖ ∇ 2 m u ‖ 2 − ε 2 λ 1 − 2 m ‖ v ‖ 2 . (2.12)

M ( ‖ ∇ m u ‖ p p − β ε − 1 ) ( ∇ 2 m u , ∇ 2 m v ) ≥ ( μ 0 − β ε − 1 ) ( ‖ ∇ 2 m u ‖ 2 4 + ‖ ∇ 2 m v ‖ 2 ) = − β ε ( ‖ ∇ 2 m u ‖ 2 4 + ‖ ∇ 2 m v ‖ 2 ) . (2.13)

Substitute inequality (2.12)-(2.13) into Equation (2.11), we get

( G ( U ) , U ) E 0 ≥ ( ε − β ε 4 − ε 2 4 ) ‖ ∇ 2 m u ‖ 2 + ( α − ε − ε 2 λ 1 − 2 m ) ‖ v ‖ 2 + ( β − β ε − α λ 1 − 2 m ) ‖ ∇ 2 m v ‖ 2 . (2.14)

According to the assumption, we can get ε − β ε 4 − ε 2 4 > 0 , α − ε − ε 2 λ 1 − 2 m > 0 , β − β ε − α λ 1 − 2 m > 0 . Let a 1 = min { ε − β ε 4 − ε 2 4 , α − ε − ε 2 λ 1 − 2 m } , a 2 = β − β ε − α λ 1 − 2 m , so we can get

( G ( U ) , U ) E 0 ≥ a 1 ‖ U ‖ E 0 2 + a 2 ‖ ∇ 2 m v ‖ 2 . (2.15)

The Lemma 2.1 is proved. Then we prove the Lipschitz property and the discrete squeezing property of S ( t ) . ■

Set S ( t ) U 0 = U ( t ) = ( u ( t ) , v ( t ) ) T , where v = u t ( t ) + ε u ( t ) ; and S ( t ) V 0 = V ( t ) = ( u ^ ( t ) , v ^ ( t ) ) T , where v ^ ( t ) = u ^ t ( t ) + ε u ^ ( t ) ; let Y ( t ) = S ( t ) U 0 − S ( t ) V 0 = U ( t ) − V ( t ) = ( w ( t ) , z ( t ) ) T , where z ( t ) = w t ( t ) + ε w ( t ) , w ( t ) = u ( t ) − u ^ ( t ) , w t ( t ) = v ( t ) − v ^ ( t ) , then Y ( t ) satisfies

Y t + G ( U ) − G ( V ) − ( 0 , g ( u ) − g ( u ^ ) ) T = 0 , (2.16)

Y ( 0 ) = U 0 − V 0 . (2.17)

Lemma 2.2. (Lipschitz property). For ∀ U 0 , V 0 ∈ B k and t ≥ 0 ,

‖ S ( t ) U 0 − S ( t ) V 0 ‖ E 0 ≤ e γ t ‖ U 0 − V 0 ‖ E 0 2 . (2.18)

Proof. Taking the inner product of the Equation (2.16) with Y ( t ) in E 0 , we can get

1 2 d d t ‖ Y ( t ) ‖ 2 + ( G ( U ) − G ( V ) , Y ( t ) ) + ( g ( u ) − g ( u ^ ) , z ( t ) ) = 0. (2.19)

Similar to Lemma 2.1, we have

( G ( U ) − G ( V ) , Y ( t ) ) E 0 ≥ a 1 ‖ Y ( t ) ‖ E 0 2 + a 2 ‖ ∇ 2 m z ( t ) ‖ E 0 2 . (2.20)

By using the condition (A1) Young’s inequality Poincare’s inequality and differential mean value theorem, we get

| ( g ( u ) − g ( u ^ ) , z ( t ) ) | ≤ | g ′ ( ξ ) | ‖ w ( t ) ‖ ‖ z ( t ) ‖ ≤ c 4 λ 1 − m ‖ ∇ 2 m w ( t ) ‖ ‖ z ( t ) ‖ ≤ c 4 λ 1 − m 2 ( ‖ ∇ 2 m w ( t ) ‖ 2 + ‖ z ( t ) ‖ 2 ) = c 4 λ 1 − m 2 ‖ Y ( t ) ‖ 2 . (2.21)

Where ξ = θ + ( 1 − θ ) u ^ , 0 < θ < 1 .

Substitute inequality (2.20)-(2.21) into equation (2.19), we get

d d t ‖ Y ( t ) ‖ 2 + 2 a 1 ‖ Y ( t ) ‖ E 0 2 + 2 a 2 ‖ ∇ 2 m z ( t ) ‖ E 0 2 ≤ c 4 λ 1 − m ‖ Y ( t ) ‖ 2 . (2.22)

We can get

d d t ‖ Y ( t ) ‖ 2 ≤ c 4 λ 1 − m ‖ Y ( t ) ‖ 2 . (2.23)

According to Gronwall’s inequality, we have

‖ Y ( t ) ‖ 2 ≤ e c 4 λ 1 − m t ‖ Y ( 0 ) ‖ 2 = e γ t ‖ Y ( 0 ) ‖ 2 . (2.24)

where γ = c 4 λ 1 − m . Therefore, we get

‖ S ( T ) U 0 − S ( T ) V 0 ‖ E 0 ≤ e γ t ‖ U 0 − V 0 ‖ E 0 2 . (2.25)

The Lemma 2.2 is proved. n

Now, we define the operator − Δ : D ( − Δ ) → H 4 m , the domain of definition is D ( − Δ ) = H 2 ( Ω ) ∩ H 0 1 ( Ω ) , obviously, − Δ is an unbounded self-adjoint closed positive operator, and ( − Δ ) − 1 is compact, we find by elementary spectral theory the existence of an orthonormal basis of H consisting of eigenvectors w j of − Δ , such that:

( ( − Δ ) w j = λ j w j , j = 1 , 2 , ⋯ , 0 < λ 1 ≤ λ 2 ≤ ⋯ λ j → ∞ as j → ∞ . (2.26)

For a given integer n, 0 < n ≤ N we denote by P n the orthogonal projection of H 4 m onto the space spanned by w 1 , ⋯ , w n i.e. p = p n = H 4 m → span { w 1 , w 2 , ⋯ , w n } , let Q n = I − P n . Then we have

‖ ( − Δ ) 2 m u ‖ ≥ λ n + 1 2 m ‖ u ‖ , ∀ u ∈ Q = Q n ( H 4 m ( Ω ) ∩ H 0 1 ( Ω ) ) , (2.27)

‖ Q n u ‖ ≤ ‖ u ‖ , u ∈ H . (2.28)

where ‖ u ‖ 2 ≤ λ n + 1 − 4 m ‖ u ‖ D ( ( − Δ ) 2 m ) 2 .

Lemma 2.3. For any U 0 , V 0 ∈ B k , ∀ n 0 ∈ N * , n 0 ≤ N , Let

Q n 0 ( t ) = Q n 0 ( U ( t ) − V ( t ) ) = Q m 0 Y ( t ) = ( ω n 0 , z n 0 ) T , (2.29)

then we have

‖ Y n 0 ( t ) ‖ E 0 2 ≤ ( e − 2 a 1 t + c 2 λ n 0 + 1 − m 2 a 1 + γ e γ t ) ‖ Y ( 0 ) ‖ E 0 2 , (2.30)

Proof. Taking projection operator Q n 0 in (2.16), we have

Y n 0 t ( t ) + Q n 0 ( G ( U ) − G ( V ) ) + ( 0, Q n 0 ( g ( u ) − g ( u ^ ) ) ) T = 0. (2.31)

Taking the inner product ( ⋅ , ⋅ ) E 0 in (2.31) with Y n 0 ( t ) , we get

1 2 d d t ‖ Y n 0 ( t ) ‖ 2 + a 1 ‖ Y n 0 ( t ) ‖ 2 + a 2 ‖ ∇ 2 m z n 0 ( t ) ‖ 2 + Q n 0 ( g ( u ) − g ( u ^ ) , z n 0 ( t ) ) = 0. (2.32)

According to (A1) and Young inequality, we have

| Q n 0 ( g ( u ) − g ( u ^ ) , z n 0 ( t ) ) | ≤ | g ′ ( ξ ′ ) | ‖ w n 0 ( t ) ‖ ‖ z n 0 ( t ) ‖ ≤ c 5 λ n 0 + 1 − m ‖ ∇ 2 m w n 0 ( t ) ‖ ‖ z n 0 ( t ) ‖ ≤ c 5 λ n 0 + 1 − m 2 ( ‖ ∇ 2 m w n 0 ( t ) ‖ 2 + ‖ z n 0 ( t ) ‖ 2 ) = c 5 λ n 0 + 1 − m 2 ‖ Y n 0 ( t ) ‖ 2 . (2.33)

where ξ ′ = θ n 0 + ( 1 − θ n 0 ) u ^ , 0 < θ n 0 < 1 .

Together with (2.32)-(2.33) and Lemma 2.2, it follows

d d t ‖ Y n 0 ‖ E 0 2 + 2 a 1 ‖ Y n 0 ( t ) ‖ 2 ≤ c 5 λ n 0 + 1 − m ‖ Y n 0 ( t ) ‖ 2 = c 5 λ n 0 + 1 − m ‖ S ( t ) U 0 − S ( t ) V 0 ‖ 2 ≤ c 5 λ n 0 + 1 − m e γ t ‖ U 0 − V 0 ‖ 2 = c 5 λ n 0 + 1 − m e γ t ‖ Y ( 0 ) ‖ 2 . (2.34)

By using Gronwall’s inequality, we get

‖ Y n 0 ( t ) ‖ 2 ≤ ‖ Y ( 0 ) ‖ 2 e − 2 a 1 t + c 5 λ n 0 + 1 − m 2 a 1 + γ e γ t ‖ Y ( 0 ) ‖ 2 = ( e − 2 a 1 t + c 5 λ n 0 + 1 − m 2 a 1 + γ e γ t ) ‖ Y ( 0 ) ‖ 2 . (2.35)

The Lemma 2.3 is proved. n

Lemma 2.4. (Discrete squeezing property). For any U 0 , V 0 ∈ B k , τ * ≥ 0 , if

‖ P n 0 ( S ( τ * ) U 0 − S ( τ * ) V 0 ) ‖ E 0 ≤ ‖ ( I − P n 0 ) ( S ( τ * ) U 0 − S ( τ * ) V 0 ) ‖ E 0 , (2.36)

then

‖ S ( τ * ) U 0 − S ( τ * ) V 0 ‖ E 0 ≤ 1 8 ‖ U 0 − V 0 ‖ E 0 . (2.37)

Proof. If ‖ P n 0 ( S ( τ * ) U 0 − S ( τ * ) V 0 ) ‖ E 0 ≤ ‖ ( I − P n 0 ) ( S ( τ * ) U 0 − S ( τ * ) V 0 ) ‖ E 0 , then

‖ S ( τ * ) U 0 − S ( τ * ) V 0 ‖ E 0 2 ≤ ‖ ( I − P n 0 ) ( S ( τ * ) U 0 − S ( τ * ) V 0 ) ‖ E 0 2 + ‖ P n 0 ( S ( τ * ) U 0 − S ( τ * ) V 0 ) ‖ E 0 2 ≤ 2 ‖ ( I − P n 0 ) ( S ( τ * ) U 0 − S ( τ * ) V 0 ) ‖ E 0 2 ≤ 2 ( e − 2 a 1 τ * + c 5 λ n 0 + 1 − m 2 a 1 + γ e γ τ * ) ‖ U 0 − V 0 ‖ E 0 2 . (2.38)

Let τ * be large enough,

e − 2 a 1 τ * ≤ 1 256 . (2.39)

Also let n 0 be large enough, we get

c 5 λ n 0 + 1 − m 2 a 1 + γ e γ τ * ≤ 1 256 . (2.40)

Substitute inequality (2.39)-(2.40) into Equation (2.38), we get

‖ S ( τ * ) U 0 − S ( τ * ) V 0 ‖ E 0 2 ≤ 1 8 ‖ U 0 − V 0 ‖ E 0 2 . (2.41)

The Lemma 2.4 is proved. n

Theorem 2.3. Let (A1), (A2) be in force, assume that f ∈ H , ( u 0 , v 0 ) ∈ E 0 ( E k ) , ( k = 1 , 2 , ⋯ , 2 m ), then the semigroup S ( t ) determined by (1.1)-(1.3) possesses an ( E k , E 0 ) -exponential attractor M k on B,

M k = ∪ 0 ≤ t ≤ τ * S ( t ) ( A k ∪ ( ∪ j = 1 ∞ ∪ i = 1 ∞ S ( τ * ) j ( E ( i ) ) ) ) , (2.42)

The fractal dimension of M k satisfies

d F ( M k ) ≤ c 3 N 0 + 1. (2.43)

Proof. According to Theorem 2.1, Lemma 2.2 and Lemma 2.4, Theorem 2.2 is easily proven. n

Next, we will prove the existence of inertial manifolds when N is large enough by using graph norm transformation method.

Definition 3.1. [

1) μ k is finite dimensional Lipschitz manifold of E k ;

2) μ k is positively invariant for the semigroup { S ( t ) } t ≥ 0 , i.e. ∀ u 0 ∈ μ k , S ( t ) u 0 ⊂ μ k , ∀ t ≥ 0 ;

3) μ k attracts exponentially all the orbits of the solution, i.e. ∃ ϑ > 0 , for ∀ u ∈ E k , ∃ k > 0 , such that

dist ( S ( t ) u , u ) ≤ k ⋅ e − ϑ t , t ≥ 0. (3.1)

Lemma 3.1. Let Λ : E k → E k be an operator and assume that F ∈ C b ( E k , E k ) satisfies the Lipschitz condition

‖ F ( U ) − F ( U ) ‖ E k ≤ l F ‖ U − V ‖ E k , U , V ∈ E k . (3.2)

The operator Λ is said to satisfy the spectral gap condition relative to F, if the point spectrum of the operator Λ can be divided into two parts σ 1 and σ 2 , of which σ 1 is finite, and we have

Λ 1 = sup { Re λ | λ ∈ σ 1 } , Λ 2 = inf { Re λ | λ ∈ σ 2 } , (3.3)

and E k i = span { ω j | j ∈ σ i , i = 1 , 2 } .

Then

Λ 2 − Λ 1 > 4 l F , (3.4)

and the orthogonal decomposition

E k = E k 1 ⊕ E k 2 , (3.5)

Then P 1 : E k → E k 1 and P 2 : E k → E k 2 are both continuous orthogonal projections . The Lemma 3.1 is proved.

Lemma 3.2. Let the eigenvalues μ j ± ( j ≥ 1 ) is non-decreasing, and for m ∈ N * , there exists N ≥ m , such that μ N + 1 − and μ N − are consecutive adjacent values.

Lemma 3.3. The function g ( u ) satisfies g : H 0 k ( Ω ) → L 2 ( Ω ) which is uniformly bounded and globally Lipschitz continuous, and l is the Lipschitz coefficient.

Proof. For ∀ u 1 , u 2 ∈ H 0 k ( Ω ) , we have

‖ g ( u 1 ) − g ( u 2 ) ‖ = ‖ g ′ ( η ) ( u 1 − u 2 ) ‖ ≤ ‖ g ′ ( η ) ‖ ∞ ‖ u 1 − u 2 ‖ H k , (3.6)

where η ∈ ( u 1 , u 2 ) , From the hypothesis (A1) and the differential mean value theorem, we know

‖ g ( u 1 ) − g ( u 2 ) ‖ ≤ C 6 ‖ u 1 − u 2 ‖ H 0 k , (3.7)

Let l = C 6 , l is the Lipschitz coefficient. n

Then we prove the existence of a family of the inertial manifold of this equation, Equation (1.1) is equivalent to the following first-order evolution equation:

U t + Λ U = F ( U ) , (3.8)

where

U = ( u , v ) T = ( u , u t ) T , Λ = ( 0 − I M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m β ( − Δ ) 2 m ) , F ( U ) = ( 0 f ( x ) − g ( u ) ) ,

D ( Λ ) = { u ∈ H 4 m ( Ω ) | u ∈ H , ( − Δ ) 2 m u ∈ H 4 m ( Ω ) } × H 2 m ( Ω ) .

We consider in E k the usual graph norm, induced by the scalar product

〈 U , V 〉 E k = ( M ⋅ ∇ 2 m + k u , ∇ 2 m + k y ¯ ) + ( v , z ¯ ) . (3.9)

where U = ( u , v ) T , V = ( y , z ) T ∈ E k , and y ¯ , z ¯ respectively denote the conjugation of y and z, and v , z ∈ H 0 2 m + k ( Ω ) , u , y ∈ H 0 2 m + k ( Ω ) . Moreover, the operator Λ is monotone, indeed, for ∀ U ∈ D ( Λ ) , we have

〈 Λ U , U 〉 E k = − ( M ⋅ ∇ 2 m + k u t , ∇ 2 m + k u ¯ ) + ( M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m u t , v ¯ ) ≥ − ( M ⋅ ∇ 2 m + k u t , ∇ 2 m + k u ¯ ) + M ( ∇ 2 m + k u , ∇ 2 m + k v ) + β ( − Δ m v , − Δ m v ¯ ) ≥ β ‖ ∇ 2 m v ‖ 2 > 0. (3.10)

so that Λ is a Monotonically increasing operator and 〈 Λ U , U 〉 E k is real and nonnegative. To determine the eigenvalues of Λ , we observe that the eigenvalue equation

Λ U = λ U , U = ( u , v ) T ∈ E k (3.11)

is equivalent to the system

{ − v = λ u , M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m v = 0. (3.12)

Thus, we can get the eigenvalue problem

{ λ 2 u + M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u − β λ ( − Δ ) 2 m u = 0 , u | ∂ Ω = ( − Δ ) 2 m u | ∂ Ω = 0. (3.13)

Using ( − Δ ) k u with the first formula of (3.13) to take the inner product, and bring u j to the position of u, we can get

λ 2 ‖ ∇ k u ‖ 2 + M ( ‖ ∇ m u ‖ p p ) ‖ ∇ 2 m + k u ‖ 2 − β λ ‖ ∇ 2 m + k u ‖ 2 = 0. (3.14)

Regarding Equation (3.14) as a quadratic equation of one variable with respect to λ , for ∀ j ∈ N + and let s = ‖ ∇ m u ‖ p p , M = M ( s ) , the corresponding eigenvalues of equation (3.11) are as follows:

λ j ± = β μ j ± β 2 μ j 2 − 4 M μ j 2 . (3.15)

where μ j ( j ≥ 1 ) is the eigenvalue of ( − Δ ) 2 m in H 0 2 m ( Ω ) , and μ j = λ 1 j 2 m n . Because of β is large enough, the eigenvalue of Λ are all positive and real numbers, the corresponding eigenvalues have the form

U j ± = ( u j , − λ j ± u j ) . (3.16)

For formula (3.15), for the convenience of later use, define the following formula

‖ ∇ 2 m + k u j ‖ = μ j , ‖ ∇ k u j ‖ = 1 , ‖ ∇ − 2 m − k u j ‖ = 1 μ j , k = 1 , 2 , ⋯ , 2 m . (3.17)

Next, it will be proved that the eigenvalue of the operator Λ satisfies the spectral interval condition.

Theorem 3.1 let l is the Lipschitz constant of g ( u ) , assume μ j ≥ 4 M ( s ) β 2 , if N 1 ∈ Z + is large enough, when N ≥ N 1 , the following inequality holds

( μ N + 1 + μ N ) ( β − β 2 μ j 2 − 4 M ( s ) ) ≥ 8 l β 2 μ j 2 − 4 M ( s ) + 1. (3.18)

Then, the operator Λ satisfies the spectral gap condition of Lemma 3.1.

Proof. Because of all the eigenvalues of the operator Λ are positive real numbers, β ≥ 2 M μ j and the sequence { λ j − } j ≥ 1 and { λ j + } j ≥ 1 are monotonically

increasing. The theorem is proved in four steps below.

step 1 Since λ j ± is a non-decreasing sequence, according to Lemma 3.2, given N, so that λ N − and λ N + 1 − are consecutive adjacent eigenvalues, the eigenvalues of the operator Λ are decomposed into σ 1 and σ 2 , where σ 1 is the finite parts, which are expressed as follows

σ 1 = { λ h − , λ j + | max { λ h − , λ j + } ≤ λ N − } , (3.19)

σ 2 = { λ h + , λ j ± | λ h − ≤ λ N − ≤ min { λ h + , λ j ± } } . (3.20)

step 2 The corresponding E k is decomposed into

E k 1 = span { U h − , U j ± | λ h − , λ j + ∈ σ 1 } , (3.21)

E k 2 = span { U h + , U j + | λ h − , λ j ± ∈ σ 2 } . (3.22)

We aim at madding two orthogonal subspaces of E k and verifying the spectral gap condition (3.4) is true when Λ 1 = λ N − , Λ 2 = λ N + 1 − . Therefore, we further decompose E k 2 = E S + E R , i.e.

E S = span { U h − | λ h − ≤ λ N − ≤ λ h + } , (3.23)

E R = span { U R + | λ N − ≤ λ j ± } . (3.24)

And set E N = E k 1 ⊕ E S . Note that E k 1 and E S are finite dimensional, that λ N − ∈ E k 1 , λ N + 1 − ∈ E R , and that the reason why E k 1 is not orthogonal to E k 2 is that, while it is orthogonal to E R , E k 1 is not orthogonal to E S . We now introduce two functions Ψ : E N → R and ψ : E R → R , defined by

Φ ( U , V ) = β ( ∇ 2 m + k u , ∇ 2 m + k y ¯ ) + 2 β ( ∇ − 2 m − k z ¯ , ∇ 2 m u ) + 2 β ( ∇ − 2 m − k v , ∇ 2 m y ¯ ) + 4 ( ∇ − 2 m − k v , ∇ − 2 m − k z ) − 4 M ( ‖ ∇ m u ‖ p p ) ( ∇ k u ¯ , ∇ k y ) + ( 2 β 2 − β ) ( ∇ 2 m + k u ¯ , ∇ 2 m + k y ) . (3.25)

Ψ ( U , V ) = ( ∇ 2 m + k u , ∇ 2 m + k y ¯ ) + ( ∇ − 2 m − k z ¯ , ∇ 2 m + k u ) − ( ∇ − 2 m − k v , ∇ 2 m + k y ¯ ) − 4 M ( ‖ ∇ m u ‖ p p ) ( ∇ k u ¯ , ∇ k y ) + ( β 2 − 1 ) ( ∇ 2 m + k u ¯ , ∇ 2 m + k y ) . (3.26)

where U = ( u , v ) T , V = ( y , z ) T ∈ E N , and y ¯ , z ¯ are respectively the conjugates of y , z . We now show that Φ and Ψ are positive definite. For ∀ U = ( u , v ) ∈ E N , we have

Φ ( U , V ) = β ( ∇ 2 m + k u , ∇ 2 m + k u ¯ ) + 2 β ( ∇ − 2 m − k v ¯ , ∇ 2 m u ) + 2 β ( ∇ − 2 m − k v , ∇ 2 m u ¯ ) + 4 ( ∇ − 2 m − k v , ∇ − 2 m − k v ) − 4 M ( ‖ ∇ m u ‖ p p ) ( ∇ k u ¯ , ∇ k u ) + ( 2 β 2 − β ) ( ∇ 2 m + k u ¯ , ∇ 2 m + k u )

≥ β ‖ ∇ 2 m + k u ‖ 2 − 4 ‖ ∇ − 2 m − k v ¯ ‖ 2 − β 2 ‖ ∇ 2 m + k u ‖ 2 − 4 M ( s ) ‖ ∇ m + k u ‖ 2 + ( 2 β 2 − β ) ‖ ∇ 2 m + k u ‖ 2 + 4 ‖ ∇ − 2 m − k v ¯ ‖ 2 = β 2 ‖ ∇ 2 m + k u ‖ 2 − 4 μ 1 ‖ ∇ k u ‖ 2 ≥ ( β 2 μ j 2 − 4 M ( s ) ) ‖ ∇ k u ‖ 2 . (3.27)

When β is large enough, we conclude that Φ ( U , U ) ≥ 0 , i.e. Φ is positive definite. Similarly, for ∀ U = ( u , v ) ∈ E R , we have

Ψ ( U , V ) = ( ∇ 2 m + k u , ∇ 2 m + k u ¯ ) + ( ∇ − 2 m − k v ¯ , ∇ 2 m + k u ) − ( ∇ − 2 m − k v , ∇ 2 m + k u ¯ ) − 4 M ( ‖ ∇ m u ‖ p p ) ( ∇ k u ¯ , ∇ k u ) + ( β 2 − 1 ) ( ∇ 2 m + k u ¯ , ∇ 2 m + k u ) ≥ ‖ ∇ 2 m + k u ‖ 2 − 4 M ( s ) ‖ ∇ m + k u ‖ 2 + ( β 2 − 1 ) ‖ ∇ 2 m + k u ‖ 2 ≥ ( β 2 μ j 2 − 4 M ( s ) ) ‖ ∇ k u ‖ 2 . (3.28)

When β is large enough, we conclude that Ψ ( U , U ) ≥ 0 , i.e. Ψ is positive definite.

Thus Φ and Ψ define a scalar product, respectively on E N and E R , and we can define an equivalent scalar product in E k , by

〈 〈 U , V 〉 〉 E k = Φ ( P N U , P N V ) + Ψ ( P R U , P R V ) . (3.29)

where P N and P R are respectively the projections of E k → E N and E k → E R . Rewrite (3.29) as follows

〈 〈 U , V 〉 〉 E k = Φ ( U , V ) + Ψ ( U , V ) . (3.30)

We proceed then to show that the subspaces E k 1 and E k 2 defined in (3.21), (3.22) are orthogonal with respect to the scalar product (3.29). In fact, it is sufficient to show that E N is orthogonal to E S , in turn, this reduces to showing that 〈 〈 U h − , U h + 〉 〉 E k = 0 if U h − ∈ E N and U h + ∈ E S . Recalling (3.27) and (3.28), we immediately compute that

〈 〈 U j + , U j − 〉 〉 E k = Φ ( U j + , U j − ) = β ( ∇ 2 m + k u j , ∇ 2 m + k u ¯ j ) + 2 β ( − λ j − ∇ − 2 m − k u ¯ j , ∇ 2 m u j ) + 2 β ( − λ j + ∇ − 2 m − k u j , ∇ 2 m u ¯ j ) + 4 ( − λ j + ∇ − 2 m − k u j , − λ j − ∇ − 2 m − k u j ) − 4 M ( ‖ ∇ m u ‖ p p ) ( ∇ k u ¯ j , ∇ k u j ) + ( 2 β 2 − β ) ( ∇ 2 m + k u ¯ j , ∇ 2 m + k u j )

= β ‖ ∇ 2 m + k u j ‖ 2 − 2 β ( λ j − + λ j + ) ‖ ∇ − k u j ‖ 2 + 4 λ j − λ j + ‖ ∇ − 2 m − k u j ‖ 2 − 4 M ( s ) ‖ ∇ k u j ‖ 2 + ( 2 β 2 − β ) ‖ ∇ 2 m + k u j ‖ 2 = 2 β 2 μ j − 2 β ( λ j − + λ j + ) + 4 λ j − λ j + 1 μ j − 4 M ( s ) . (3.31)

According to (3.15), we have

λ j − + λ j + = β μ j . (3.32)

λ j − λ j + = M μ j . (3.33)

Therefore

〈 〈 U j + , U j − 〉 〉 E k = Φ ( U j + , U j − ) = 0. (3.34)

step 3 Further, we estimate the Lipschitz constant l F of F ( U ) = ( 0 , f ( x ) − g ( u ) ) T , according to Lemma 3.3 we can get g : H 0 2 m ( Ω ) → L 2 ( Ω ) is uniformly bounded and globally Lipschitz continuous. For ∀ U ( u , v ) T ∈ E k , U i = ( u i , v i ) T ∈ P i U ( i = 1 , 2 ) , we have

‖ U ‖ E k 2 = Φ ( P 1 U , P 1 U ) + Ψ ( P 2 U , P 2 U ) ≥ ( β 2 μ j 2 − 4 M ( s ) ) ‖ ∇ k P 1 u ‖ 2 + ( β 2 λ 1 m − 4 μ 1 ) ‖ ∇ k P 2 u ‖ 2 ≥ ( β 2 μ j 2 − 4 M ( s ) ) ‖ ∇ k u ‖ 2 . (3.35)

Given U = ( u , v ) T , V = ( u ˜ , v ˜ ) T = ( y , z ) T ∈ E k , we have

‖ F ( U ) − F ( V ) ‖ E k = ‖ g ( u ) − g ( u ˜ ) ‖ ≤ l ‖ u − u ˜ ‖ ≤ 1 β 2 μ j 2 − 4 M ( s ) ‖ U − V ‖ E k . (3.36)

Thus, we have

l F ≤ 1 β 2 μ j 2 − 4 M ( s ) . (3.37)

step 4 Now, we will show the spectral gap condition (3.4) holds.

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

Λ 2 − Λ 1 = λ N + 1 − − λ N − = β 2 ( μ N + 1 − μ N ) + 1 2 ( R ( N ) − R ( N + 1 ) ) . (3.38)

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

There exists N 1 ≥ 0 , such that for ∀ N ≥ N 1 , R 1 ( N ) = 1 − β 2 β 2 μ j 2 − 4 M ( s ) − 4 M β 2 μ j 2 − 4 M ( s ) . We can get

R ( N ) − R ( N + 1 ) + β 2 μ j 2 − 4 M ( s ) ( μ N + 1 − μ N ) = β 2 μ j 2 − 4 M ( s ) ( μ N + 1 R 1 ( N + 1 ) − μ N R 1 ( N ) ) , (3.39)

According to assumption (A2), we can easily see that

l i m N → + ∞ ( R ( N ) − R ( N + 1 ) + β 2 μ j 2 − 4 M ( s ) ( μ N + 1 − μ N ) ) = 0, (3.40)

Then according to (3.18) and (3.37)-(3.40), we have

Λ 2 − Λ 1 ≥ 1 2 ( ( μ N + 1 − μ N ) ( β − β 2 μ j 2 − 4 M ( s ) ) − 1 ) ≥ 4 l β 2 μ j 2 − 4 M ( s ) ≥ 4 l F . (3.41)

The Theorem 3.1 is proved.

Theorem 3.2. Under the conclusion of Theorem 3.1, the problem (1.1)-(1.3) exists a family of inertial manifolds μ k in E k

μ k = graph ( m ) : = { ζ k + γ ( ζ k ) : ζ k ∈ E k 1 } (3.42)

where E k 1 , E k 2 defined in (3.21)-(3.22), and χ : E k 1 → E k 2 is Lipschitz continuous function. n

The authors declare no conflicts of interest regarding the publication of this paper.

Lin, G.G. and Yang, L.J. (2021) A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations. Journal of Applied Mathematics and Physics, 9, 2399-2413. https://doi.org/10.4236/jamp.2021.910152