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 APM  Vol.11 No.4 , April 2021
Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations
Abstract: In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

1. Introduction

The objective of this paper is to study the following initial boundary value problem of the generalized Kirchhoff equation

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 ) will be given later.

In 1883, German physicist G. Kirchhoff [1] first introduced the following model to study the free vibration of elastic strings

ρ h 2 u 2 t + δ u t = P 0 + E h 2 L ( 0 L | u x | 2 d x ) + f ( x ) , 0 < x < L , t > 0.

where the time variable is t, the elastic modulus is E, h is the cross-sectional area, L is the length of the string, ρ is the mass density, P 0 is the initial axial tension, δ is the resistance coefficient, f is the external force term, u = u ( x , t ) is the lateral displacement at the space coordinate x and the time t.

Since the 1980s, with the progress of science and technology and the continuous development of mathematical physics equations and Kirchhoff equation has been widely used, especially in measuring bridge vibration of engineering physics, so more and more scholars begin to pay close attention to and carries on the thorough study of Kirchhoff equation and a series of theories and research results in recent years, including the existence and uniqueness of the global solutions, global attractor and Hausdorff dimension and fractal dimension, the existence of random attractor, energy decay and blow-up of the solution, exponential attractor and inertial manifolds, etc. Among them, scholars have done a lot of research on the existence of global attractors for Kirchhoff-type equations with strong dissipation terms, the theoretical basis and research results can be found in the references ( [2] - [9] ).

In reference [10], Guoguang Lin, Yunlong Gao studied the existence and uniqueness of global solutions of a class of generalized Kirchhoff-type equations with nonlinear strong damping and their global attractors

u t t + ( Δ ) m u t + ( α + β ( D m u 2 ) ) ( Δ ) m u + g ( u ) = f ( x ) , ( x , t ) Ω × [ 0 , + )

u ( x , t ) = 0 , i u v i = 0 , i = 1 , 2 , , m 1 , x Ω , t [ 0 , + )

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω

By assuming the nonlinear source terms g(u), the author verifies the appropriateness of the solution and proves the existence of the global attractor.

Recently, Lin Guoguang and Guan Liping [11] studied the global attractor of a high-order Kirchhoff-type equation with a strong nonlinear damping term and finite dimensional estimation of its Hausdorff dimension and Fractal dimension

u t t + M ( D m u 2 ) ( Δ ) m u + β ( Δ ) m u t + Δ g ( u ) = f ( x )

where m > 1 , Ω is a bounded domain with smooth homogeneous Dirichlet boundary Ω R n , f ( x ) denotes the external force, Δ g ( u ) denotes second order nonlinear source term, M is a general function, β > 0 , β ( Δ ) m u t is strong damping term. For more significant research results about the global attractor and its dimension estimation of Kirchhoff equation, please refer to the literature ( [12] [13] [14] [15] ).

In this paper, on the basis of literature [11], the rigid term D m u 2 is extended to D m u p p , the existence and uniqueness of global solutions of generalized nonlinear Kirchhoff-type equations are proved, and the existence of global attractors and their finite Hausdorff dimension and Fractal dimension of problems (1)-(3) are discussed.

For convenience, define the following spaces and notations

H = L 2 ( Ω ) , H 0 m ( Ω ) = H 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 ) , f ( x ) L 2 ( Ω ) , E k = H 0 2 m + k ( Ω ) × H 0 k ( Ω ) , ( k = 1 , 2 , , 2 m ) .

( , ) 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 , = u L 2 ( Ω ) , P = u L P ( Ω ) , = u L ( Ω ) .

Let’s call A k ( k = 1 , 2 , , 2 m ) is the weak global attractor of E 0 to E k , B o k is a bounded absorption set in E k , C i > 0 ( i = 0 , 1 , 2 , ) is constant.

Assume that the nonlinear source term g(u) in Equation (1.1) satisfies the following conditions

(A1) g ( u ) C ( R ) ;

(A2) J ( u ) = G ( u ) d x , where G ( u ) = g ( u ) u t ;

(A3) J ( u ) μ 4 2 m + k u 2 C .

The Kirchhoff-type stress term satisfies the following conditions

(A4) M ( s ) C 2 ( [ 0 , + ) , R + ) ;

(A5) ε + 1 = μ 0 M ( s ) μ 1 , where μ = { μ 0 , d d t m u 2 0 μ 1 , d d t m u 2 < 0

where μ , μ 0 , μ 1 are constant, and 0 < ε < min { 1 + 2 β λ 1 2 m 1 2 , μ 0 + μ 0 2 λ 1 2 m λ 1 2 m , 1 + β λ 1 2 m 1 2 , 2 μ 0 β + λ 1 2 m } , λ 1 is the first eigenvalue of Δ with homogeneous Dirichlet boundary conditions on Ω .

2. A Priori Estimate of Smooth Solution

Lemma 1. Assume that the nonlinear terms g(u), M(s) satisfies assumptions (A1)-(A5), and f L 2 ( Ω ) , ( u 0 , v 0 ) E 0 = H 2 m ( Ω ) × L 2 ( Ω ) , then the initial boundary value problem (1.1)-(1.3) has a smooth solution ( u , v ) E 0 and v L 2 ( 0, T ; H 2 m ( Ω ) )

( u , v ) E 0 2 = 2 m u 2 + v 2 Y ( 0 ) e α 1 t + C 1 α 1 ( 1 e α 1 t ) . (2.1)

where v = u t + ε u , α 1 = min { a 1 , a 2 μ , 1 } , Y ( 0 ) = v 0 2 + μ 2 m u 0 2 + 2 J ( u 0 ) , so there’s a non-negative real number R 0 = 2 C 1 α 1 and t 1 = 1 α 1 ln ( α 1 Y ( 0 ) C 1 ) , and 0 T 2 m v 2 d t C , such that

( u , v ) E 0 2 = 2 m u 2 + v 2 R 0 2 , ( t > t 1 ) . (2.2)

Proof. Set v = u t + ε u , take the inner product of both sides of Equation (1.1) with v in H, we obtain

( u t t + M ( m u p p ) ( Δ ) 2 m u + β ( Δ ) 2 m u t + g ( u ) , v ) = ( f ( x ) , v ) . (2.3)

( u t t , v ) = ( v t ε v + ε 2 u , v ) 1 2 d d t v 2 2 ε + ε 2 2 v 2 ε 2 2 λ 1 2 m 2 m u 2 . (2.4)

( M ( m u p p ) ( Δ ) 2 m u , v ) = ( M ( m u p p ) 2 m u , 2 m ( u t + ε u ) ) = M ( m u p p ) 2 d d t 2 m u 2 + ε M ( m u p p ) 2 m u 2 μ 2 d d t 2 m u 2 + ε μ 0 2 m u 2 . (2.5)

By using the Poincare’s inequality, we obtain

( β ( Δ ) 2 m u t , v ) = β 2 m v 2 ( β ε ( Δ ) 2 m u , v ) β 2 2 m v 2 + β λ 1 2 m 2 v 2 1 2 2 m u 2 β 2 ε 2 2 2 m v 2 . (2.6)

The following estimation can be obtained from hypothesis (A2)

( g ( u ) , v ) = ( g ( u ) , u t ) + ε ( g ( u ) , u ) = d d t G ( u ) d x + ε ( g ( u ) , u ) d d t G ( u ) d x + ε 2 G ( u ) d x d d t J ( u ) + J ( u ) . (2.7)

By using the weighted Young’s inequality, we obtain

( f ( x ) , v ) f ( x ) v 1 2 ε 2 f ( x ) 2 + ε 2 2 v 2 . (2.8)

Substitute inequality (2.4)-(2.8) into Equation (2.3), therefore

d d t [ v 2 + μ 2 m u 2 + 2 J ( u ) ] + ( β λ 1 2 m 2 ε 2 ε 2 ) v 2 + ( β β 2 ε 2 ) 2 m v 2 + ( 2 ε μ 0 ε 2 λ 1 2 m 1 ) 2 m u 2 + 2 J ( u ) f ( x ) 2 ε 2 + C 0 . (2.9)

Let a 1 = β λ 1 2 m 2 ε 2 ε 2 0 , a 2 = 2 ε μ 0 ε 2 λ 1 2 m 1 0 , β β 2 ε 2 0 , and let α 1 = min { a 1 , a 2 μ , 1 } , C 1 = f ( x ) 2 ε 2 + C 0 , then Equation (2.9) can be reduced to

d d t Y ( t ) + α 1 Y ( t ) + ( β β 2 ε 2 ) 2 m v 2 C 1 . (2.10)

According to hypothesis (A3)

( v 2 + 2 m u 2 ) min ( 1 , μ 2 ) v 2 + μ 2 2 m u 2 + ( μ 2 2 m u 2 + 2 J ( u ) ) C . (2.11)

Then

Y ( t ) = v 2 + μ 2 m u 2 + 2 J ( u ) > 0. (2.12)

By using the Gronwall’s inequality, we get

( u , v ) E 0 2 = 2 m u 2 + v 2 Y ( 0 ) e α 1 t + C 1 α 1 ( 1 e α 1 t ) . (2.13)

And

l i m t ¯ ( u , v ) E 0 2 C 1 α 1 . (2.14)

So, there are constants R 0 = 2 C 1 α 1 and t 1 = 1 α 1 ln ( α 1 Y ( 0 ) C 1 ) > 0 , we obtain

( u , v ) E 0 2 = 2 m u 2 + v 2 R 0 2 , ( t > t 1 ) (2.15)

The Lemma 1 is proved.

Lemma 2. Assume that the nonlinear terms g(u), M(s) satisfies assumptions (A1)-(A5), and f H k ( Ω ) , ( u 0 , v 0 ) E k = H 2 m + k ( Ω ) × H k ( Ω ) , ( k = 1 , 2 , , 2 m ) , then the initial boundary value problem (1.1)-(1.3) has a smooth solution u ( x , t ) , u t ( x , t ) + ε u ( x , t ) = v ( x , t ) satisfy

( u , v ) E k 2 = 2 m + k u 2 + k v 2 C 2 e α 2 t + C 2 α 2 ( 1 e α 2 t ) . (2.16)

where v = u t + ε u , α 2 = min { b 1 , b 2 μ } , Z ( 0 ) = k v ( 0 ) 2 + μ 2 m + k u ( 0 ) 2 , so there are non-negative real number R 1 = 2 C 2 α 2 and t 2 = 1 α 2 ln ( α 2 Z ( 0 ) C 2 ) > 0 , such that

( u , v ) E k 2 = 2 m + k u 2 + k v 2 R 1 2 , ( t > t 2 ) . (2.17)

Proof. Set ( Δ ) k v = ( Δ ) k u t + ε ( Δ ) k u , take the inner product of both sides of equation(1.1) with ( Δ ) k v in H, we obtain

( u t t + M ( m u p p ) ( Δ ) 2 m u + β ( Δ ) 2 m u t + g ( u ) , ( Δ ) k v ) = ( f ( x ) , ( Δ ) k v ) . (2.18)

( u t t , ( Δ ) k v ) = ( v t ε v + ε 2 u , ( Δ ) k v ) 1 2 d d t k v 2 2 ε + ε 2 2 k v 2 ε 2 2 λ 1 2 m 2 m + k u 2 . (2.19)

According to hypothesis (A5), and use a proof method similar to lemma 1, we can get

( M ( m u p p ) ( Δ ) 2 m u , ( Δ ) k v ) = M ( m u p p ) 1 2 d d t 2 m + k u 2 + ε M ( m u p p ) 2 m + k u 2 μ 2 d d t 2 m + k u 2 + ε μ 0 2 m + k u 2 . (2.20)

By using Poincare’s inequality and Young’s inequality, we have

( β ( Δ ) 2 m u t , ( Δ ) k v ) = β ( ( Δ ) 2 m v ε ( Δ ) 2 m u , ( Δ ) k v ) β 2 2 m + k v 2 + β λ 1 2 m 2 k v 2 1 2 2 m + k u 2 β 2 ε 2 2 2 m + k v 2 . (2.21)

( g ( u ) , ( Δ ) k v ) = g ( u ) 2 k v β 8 2 m + k v 2 C 3 . (2.22)

where C 3 = 2 g ( u ) β λ 1 2 m k .

( f ( x ) , ( Δ ) k v ) = ( k f ( x ) , k v ) k f 2 2 ε 2 + ε 2 2 k v 2 . (2.23)

Substitute inequality (2.19)-(2.23) to (2.18), therefore

d d t [ k v 2 + μ 2 m + k u 2 ] + ( β λ 1 2 m 2 ε 2 ε 2 ) k v 2 + ( 2 ε μ 0 ε 2 λ 1 2 m 1 ) 2 m + k u 2 + ( 3 β 4 β 2 ε 2 ) 2 m + k v 2 k f ( x ) 2 ε 2 + C 3 . (2.24)

Obviously, there is a non-negative ε , such that b 1 = β λ 1 2 m 2 ε 2 ε 2 0 , b 2 = 2 ε μ 0 ε 2 λ 1 2 m 1 0 , 3 β 4 β 2 ε 2 0 , let α 2 = min { b 1 , b 2 μ } , C 2 = k f ( x ) 2 ε 2 + C 3 , then (2.24) can reduce to

d d t Z ( t ) + α 2 Z ( t ) + ( 3 β 4 β 2 ε 2 ) 2 m + k v 2 C 2 . (2.25)

where

Z ( t ) = k v 2 + μ 2 m + k u 2 > 0. (2.26)

By using Gronwall’s inequality, we can get

( u , v ) E k 2 = k v 2 + μ 2 m + k u 2 Z ( 0 ) e α 2 t + C 2 α 2 ( 1 e α 2 t ) . (2.27)

and

l i m t ¯ ( u , v ) E k 2 C 2 α 2 . (2.28)

So, there are constants R 1 = 2 C 2 α 2 and t 2 = 1 α 2 ln ( α 2 Z ( 0 ) C 2 ) > 0 , we obtain

( u , v ) E k 2 = 2 m + k u 2 + k v 2 R 1 2 , ( t > t 2 ) . (2.29)

The Lemma 2 is proved.

3. Existence and Uniqueness of Solutions

Theorem 1 Assume that the nonlinear terms g(u), M(s) satisfies (A1)-(A5), f H , ( u 0 , v 0 ) E 0 , then the initial boundary (1)-(3) exists a unique smooth solution ( u , v ) L ( [ 0 , + ) ; E 0 ) . Proof. To prove the existence, the application of Calerkin’s method is divided into the following three steps.

1) Approximate solution.

Suppose the eigenvector ω j of ( ( Δ ) 2 m ) ω j = λ j 2 m ω j generates an orthonormal basis for H 2 m , where λ j is the eigenvalue of Δ with homogeneous

Dirichlet boundary on Ω , the k-order approximation u k ( t ) is defined as follows: u k ( t ) S p a n [ ω 1 , ω 2 , , ω k ] , u k ( t ) = j = 1 k g j k ( t ) ω j .

( u k ( t ) , ω j ) + M ( m u k ( t ) p p ) ( 2 m u k ( t ) , 2 m ω j ) + β ( 2 m u ( t ) , 2 m ω j ) + ( g ( u k ) , ω j ) = ( f ( x ) , ω j ) (3.1)

where j = 1 , 2 , , k .

u k ( 0 ) = u 0 k , u k ( 0 ) = u 1 k and.

In H, , the system of ordinary differential equations with respect to can be determined on the interval, we need to prove that.

2) Prior estimate.

According to the conclusion and proof method of lemma 1, is uniformly bounded on, that is

(3.2)

(3.3)

(3.4)

where R is a constant independent of k. According to lemma2, we get, therefore, inequality (3.2)-(3.4) indicate is bounded in, is bounded in, in fact, we can get and.

3) Limit process.

According to Danford-Pttes throrem, space is conjugated to space, space is conjugated to space, select the subsequence from the sequence, such that

is weak * convergence in;

is weak * convergence in;

is weak convergence in.

By the Rellich-Kohdrachov compact embedding theorem, is compact embedded in and, then converges strongly almost everywhere in and.

converges in;

is weak * convergence in;

is weak * convergence in.

Because of, so

is weak * convergence in.

is converges in.

From Equation (3.1), the following formula can be derived this is true for.

By the density of the base, then for, the following equation is established

Then is weak convergence in, and in, we establish

;

;

So is satisfied for all j, so that existence can be proved.

Then prove the uniqueness of the solution.

Assume and are solutions of Equation (1.1), let, and substitute and into this equation, we can obtain

(3.5)

Take the inner product of both sides of Equation (3.5) with in H, then

(3.6)

According to Sobolev embedding theorem, , there exists constant, we have

(3.7)

where

(3.8)

By lemma 1, lemma 2, differential mean value theorem and Young’s inequality, we can obtain

(3.9)

where.

According to the hypothesis (A1), we get

(3.10)

Substitute inequality (3.9), (3.10) to (3.6), we get

(3.11)

By using Gronwall’s inequality, we get

(3.12)

where.

So we can get

(3.13)

Then, so, then the uniqueness of solutions is proved.

The theorem 1 is proved completely.

4. The Existence of the Family of Global Attractor

Theorem 2 [16] Assume that E is a Banach space, is a semigroup operator onE, and, , , where I is unit operator, suppose satisfies the following conditions:

1) Semigroup S(t) is uniformly bounded in E, that is for all, it exists a constant C(r), such that,

2) It exists a bounded absorbing set, that is for all, it exists a constant,

3) is completely continuous operator.

Thus there is a compact global attractor for the semigroup operator.

If the Banach space E is changed to Hilbert space in theorem 2, the existence theorem of the family of the global attractors can be obtained.

Theorem 3 Under the assumption of lemma 1, lemma 2 and theorem 1, problem (1.1)-(1.3) exist a family of the global attractors

where, is a bounded absorbing set in, that is exists a compact set satisfies the following conditions:

1)

2), (is bounded set), where

S(t) is the solution semigroup generated by problem (1.1)-(1.3).

Proof. We need to prove the three conditions of theorem 2, according to theorem 1, lemma 2, we know the problem (1.1)-(1.3) could generate the solution semigroup,.

1) According to lemma 1 and lemma 2, we know that for any bounded set, and

where, , it indicates is uniformly bounded in.

2) According to lemma 2, , for

Thus, semigroup S(t) exists bounded absorbing set.

3) is compact embedded, it indicates the bounded set in is the compact set in, so semigroup operator S(t) is completely continuous. Furthermore we can get semigroup operator S(t) exists a compact family of the

global attractor.

Theorem 3 is proved.

5. Estimation of the Dimension of the Family of Global Attractors

Let’s consider the linearization problem of (1.1)-(1.3)

(5.1)

(5.2)

(5.3)

where, is the solution of the problem (1.1)-(1.3) which take as the initial value., , it can be proved that for any, linearized initial boundary value problem (5.1)-(5.3) have unique solution.

Theorem 4. The Frechet derivative of mapping is the linear operator on, thus for any, mapping is Frechet differentiable on, and is the solution to Equations (5.1)-(5.3). Proof. Let,

, and, define, , c is a constant. We can get the Lipschitz property of S(t) on the bounded set, that is

(5.4)

Let is the solution of problem (1.1)-(1.3), we have

(5.5)

(5.6)

Let, we obtained

(5.7)

where.

(5.8)

(5.9)

Let, then

(5.10)

where, then

(5.11)

Take the inner product of (5.5) with, we get

(5.12)

And we get

(5.13)

Let, according to lemma 1, lemma 2, differential mean value theorem, interpolation inequality and Poincare’s inequality, we obtained

(5.14)

(5.15)

(5.16)

Sum up (5.14)-(5.16), we get

(5.17)

Then substitute inequality (5.14)-(5.17) to inequality (5.12), and by using Young’s inequality, Poincare’s inequality can obtained

(5.18)

According to Gronwall’s inequality, we get

(5.19)

According to (5.19), we can get, the following is established

(5.20)

The theorem 4 is proved.

The following will show that the family of the global attractor have finite Hausdorff dimension and Fractal dimension.

Theorem 5 In theorem 3, the family of the global attractor of equation (1.1)-(1.3) have finite Hausdorff dimension and Fractal dimension, and.

Proof we can write Equation (1.1) as

(5.21)

Suppose, , , , is a isomorphic mapping, and Equation (5.21) can write as

(5.22)

where.

(5.23)

(5.24)

Assume is Frechet differential, then linearize Equation (5.23), we get

(5.25)

where

, and, U is the solution of Equation (5.22).

For a fixed, assume are elements of, and suppose are solutions of linear Equation (5.1), and corresponding initial values are, so we have

(5.26)

where denotes the exterior product, denotes the trace of the operator, is the orthogonal projection from space to.

For a given time, assume is orthonormal basis of.

Define the inner product of

To sum up, we get

(5.27)

And

(5.28)

By using Holder inequality, Young’s inequality and Poincare inequality

(5.29)

where

Let, then we get

(5.30)

Because of is orthonormal basis, thus

(5.31)

(5.32)

There exists, and, we have

(5.33)

So

(5.34)

Then assume, A is bounded absorbing set of, , , let

(5.35)

(5.36)

By the inequality (5.34)-(5.36), we can get

(5.37)

Thus, the Lyapunov exponent of is uniformly bounded, that is

(5.38)

So, there exists a, such that

(5.39)

where is the eigenvalue of A, and, then

(5.40)

So

(5.41)

Thus, we can get the conclusion.

Cite this paper: Lin, G. and Yang, L. (2021) Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations. Advances in Pure Mathematics, 11, 317-333. doi: 10.4236/apm.2021.114020.
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