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The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping

  • Penghui Lv EMAIL logo , Guoguang Lin and Xiaojun Lv
Published/Copyright: March 28, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

The Kirchhoff model is derived from the vibration problem of stretchable strings. This article focuses on the long-time dynamics of a class of higher-order coupled Kirchhoff systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and the Faedo-Galerkin method. Subsequently, the family of global attractors of these problems is proved using the compactness theorem. In this article, we systematically propose the definition and proof process of the family of global attractors and enrich the related conclusions of higher-order coupled Kirchhoff models. The conclusions lay a theoretical foundation for future practical applications.

Keywords: higher-order coupled Kirchhoff system; nonlinear strong damping; global attractor family
MSC 2010: 35B41; 35G31

1 Introduction

In this study, we consider the dynamic behavior of the following higher-order coupled Kirchhoff models in a bounded smooth domain Ω ⊂ R n :

(1) u t t + N 1 ( ‖ ∇ m 1 u ‖ 2 ) ( − Δ ) m 1 u t + M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( − Δ ) m 1 u + g 1 ( u , v ) = f 1 ( x ) , v t t + N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ( − Δ ) 2 m 2 v t + M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( − Δ ) 2 m 2 v + g 2 ( u , v ) = f 2 ( x ) ,

under the following boundary conditions:

(2) u ( x ) = 0 , ∂ i u ∂ n i = 0 , i = 1 , … , m 1 − 1 , m 1 > 1 v ( x ) = 0 , ∂ j v ∂ n j = 0 , j = 1 , … , 2 m 2 − 1 , m 2 > 1 ,

and the following initial conditions:

(3) u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , v ( x , 0 ) = v 0 ( x ) , v t ( x , 0 ) = v 1 ( x ) , x ∈ Ω ,

where Δ is the Laplace operator, N 1 , N 2 and M 1 , M 2 are scalar functions specified later, g 1 , g 2 are the given source terms, and f 1 , f 2 are the given functions.

(1) is a set of important generalized higher-order quasi-linear wave equations. The proposed equation in this article originated from Kirchhoff’s vibration problem of stretchable strings in 1883:

(4) ρ h ∂ 2 u ∂ t 2 = p 0 + E h 2 L ∫ 0 L ∂ u ∂ x 2 d x ∂ 2 u ∂ t 2 ,

where 0 < x < L , t ≥ 0 , u = u ( x , t ) is the lateral displacement at space coordinate x and time coordinate t , E represents the Young’s modulus, ρ represents the mass density, h represents the cross-sectional area, L represents the length, and p 0 represents the axial tension of the accident. The long-time behavior of various forms of Kirchhoff equations have attracted the attention of many scholars in recent decades, and abundant research results have been produced [1–13].

Chueshov [1] studied the well-posedness and long-time dynamical behavior of the following Kirchhoff equation with a nonlinear strong damping term:

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

Lin et al. [2] studied the global dynamics of the following generalized nonlinear Kirchhoff-Boussinesq equations with a strong damping:

(6) u t t + α u t − β Δ u t + Δ 2 u = div ( g ( ∣ ∇ u ∣ 2 ) ∇ u ) + Δ h ( u ) + f ( x ) .

This article proved that the semi-group conformed to the squeezing property and demonstrated the existence of the exponential attractor of the system. Then, the spectral interval theory was used to prove that the system had an inertial manifold.

Ghisi and Gobbino [3] studied the global and local existence of solutions to the following Kirchhoff model with strong damping:

(7) u t t ( t ) + 2 δ A σ u t ( t ) + M ( ∣ A 1 / 2 u ( t ) ∣ 2 ) A u ( t ) = 0 .

Nakao [4] proved the initial-boundary value problem of the quasi-linear Kirchhoff-type wave equation with standard dissipation u t :

(8) u t t − ( 1 + ‖ ∇ u ( t ) ‖ 2 2 ) Δ u + u t + g ( x , u ) = f ( x ) .

With the advance of research, scholars began to turn their attention to the dynamics of the higher-order Kirchhoff equations. Ye and Tao [14] studied the initial-boundary value problem of the following kind of higher-order Kirchhoff-type equation with a nonlinear dissipation term:

(9) u t t + Φ ( ‖ D m u ‖ 2 ) ( − Δ ) m u + a ∣ u t ∣ q − 2 u t = b ∣ u ∣ r − 2 u .

Lin and Zhu [15] studied the initial and boundary value problems of the following nonlinear nonlocal higher-order Kirchhoff-type equations:

(10) u t t + M ( ‖ D m u ‖ 2 ) ( − Δ ) m u + β ( − Δ ) m u t + g ( x , u t ) = f ( x ) .

This study demonstrated the existence and uniqueness of the solution, proved the existence of a global attractor family of the problem through the compact method, and obtained the finite Hausdorff and Fractal dimensions.

Originated from physics, system coupling is a measure where two entities depend on each other. With suitable conditions or parameters, a connected system is coupled, and the potential energy of the system can enable the combination of structural functions of different systems and generate new functions. As a mathematical equation derived from physics, the Kirchhoff model is favorable for considering coupled system. Scholars gradually considered the dynamics of coupled Kirchhoff equations. For example, Wang and Zhang [16] studied the long-time dynamics problem of a class of coupled beam equations with strong damping under nonlinear boundary conditions. Lin and Zhang [17] studied the initial-boundary value problem of the following Kirchhoff coupling group with strong damping and source terms:

(11) u t t − β Δ u t − M ( ‖ ∇ u ‖ 2 + ‖ ∇ v ‖ 2 ) Δ u + g 1 ( u , v ) = f 1 ( x ) , v t t − β Δ v t − M ( ‖ ∇ u ‖ 2 + ‖ ∇ v ‖ 2 ) Δ v + g 2 ( u , v ) = f 2 ( x ) .

The finite Hausdorff dimension of the global attractor can be obtained in [17].

In recent years, Lin et al. [18–20] focused on the dynamics of a class of higher-order coupled Kirchhoff equations and obtained a series of ideal results.

At present, few articles focus on the higher-order coupled Kirchhoff problems, and the problem of higher-order beam-plate coupled with nonlinear strong damping has not been studied. The main difficulty lies in the estimation and processing of the harmonic term and the nonlinear damping term and the nonlinear damping when proving the uniqueness. Therefore, under reasonable assumptions, this article overcomes these difficulties by using Holder’s inequality, Young’s inequality, Poincare inequality, and Gagliardo-Nirenberg inequality and obtains the global solution of the problem and the family of global attractors. This study could refine the definition and existence theorem of the family of global attractors. The conclusions could fill the gap of the family of global attractors of higher-order coupled models (regardless of whether m 1 is equal to m 2 ) and lay the foundation for subsequent engineering applications.

This article is organized as follows. Section 2 presents the fundamentals for this work. Section 3 states and proves the main results. Finally, conclusions of this article are presented in Section 4.

2 Preparatory knowledge

In this article, ‖ ⋅ ‖ and ( ⋅ , ⋅ ) denote the norm and the inner product in H = L 2 ( Ω ) . Let H 0 1 = D ( ( − Δ ) 1 2 ) be the scale of the Hilbert space generated by the Laplacian with Dirichlet boundary condition on H and endowed with standard inner product and norm, respectively, ( ⋅ , ⋅ ) H 0 1 = ( ( − Δ ) 1 2 ⋅ , ( − Δ ) 1 2 ⋅ ) and ‖ ⋅ ‖ H 0 1 = ‖ ( − Δ ) 1 2 ⋅ ‖ . The main goal is to study the well-posedness and long-time dynamics of problems (1) to (3) under the following set of hypotheses:

(A1). M ( s ) is a continuous function on intervals [ 0 , + ∞ ) , M ( s ) ∈ C 1 ( R + ) , and

  1. M ′ ( s ) ≥ 0 ,

  2. M ( 0 ) ≡ M 0 > 0 .

(A2). For any u , v ∈ H , let

J ( u , v ) = ∫ Ω [ G 1 ( u , v ) + G 2 ( u , v ) ] d x ,

where G 1 ( u , v ) = ∫ 0 u g 1 ( s , v ) d s , G 2 ( u , v ) = ∫ 0 v g 2 ( u , s ) d s , then for any μ ≥ 0 , there exists C 1 ≥ 0 , C μ ≥ 0 , C μ ′ ≥ 0 that

G 1 ( u , v ) + G 2 ( u , v ) − C 1 J ( u , v ) + μ ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ≥ − C μ , J ( u , v ) + μ ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ≥ − C μ ′ .

(A3). g j ( u , v ) ( j = 1 , 2 ) ∈ C 1 ( R ) , and

∣ g j ( u , v ) ∣ ≤ C 2 ( 1 + ∣ u ∣ p j + ∣ v ∣ q j ) ; ∣ g j u ( u , v ) ∣ ≤ C 3 ( 1 + ∣ u ∣ p j − 1 + ∣ v ∣ q j ) ; ∣ g j v ( u , v ) ∣ ≤ C 4 ( 1 + ∣ u ∣ p j + ∣ v ∣ q j − 1 ) .

Specifically, when n = 1 , 2 , 1 ≤ p j ( q j ) ; when 3 ≤ n ≤ 4 m , 1 ≤ p j ( q j ) ≤ 2 m n − 2 m , where m = min { m 1 , m 2 } .

(A4). N j ( s j ) ≥ N j 0 and N j 0 ( j = 1 , 2 ) are positive constants, and ρ > 0 . Thus, M ( s 1 + s 2 ) − ρ N 1 ( s 1 ) − ρ N 2 ( s 2 ) > 0 .

Then, the research phase space of this study is obtained:

V 0 = H , V k = H k ( Ω ) ∩ H 0 1 , X 0 × 0 = V m 1 ( Ω ) × H × V 2 m 2 × H , X k 1 × k 2 = V m 1 + k 1 ( Ω ) × V k 1 ( Ω ) × V 2 m 2 + 2 k 2 × V 2 k 2 ( Ω ) , k 1 = 0 , 1 , 2 , … , m 1 , k 2 = 0 , 1 , 2 , … , m 2 ,

and the norms of the corresponding spaces are as follows:

‖ ( u , y 1 , v , y 2 ) ‖ X k 1 × k 2 2 = ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 .

Meanwhile, the general form of the Poincare inequality is as follows: λ 1 ‖ ∇ r u ‖ 2 ≤ ‖ ∇ r + 1 u ‖ 2 , where λ 1 is the first eigenvalue of − Δ with a homogeneous Dirichlet boundary on Ω . In this article, C i is a constant, and C ( ⋅ ) is a constant depending on the parameters in parentheses.

Lemma 1

[21] Let y : R + → R + be an absolutely continuous positive function on [ 0 , + ∞ ) , which satisfies the following differential inequality for some δ > 0 :

d d t y ( t ) + 2 δ y ( t ) ≤ g ( t ) y ( t ) + K , t > 0 ,

where K ≥ 0 , and a ≥ 0 if t ≥ s ≥ 0 so that ∫ s t g ( τ ) d τ ≤ δ ( t − s ) + a . Then,

y ( t ) ≤ e a y ( 0 ) e − δ t + K e a δ , t ≥ 0 .

Lemma 2

[15] Let X be a Banach space, and the continuous operator semi-group { S ( t ) } t ≥ 0 satisfies the following:

  1. The semi-group { S ( t ) } t ≥ 0 is uniformly bounded in X , i.e., for all R 0 > 0 , and there exists a positive constant C 0 ( R 0 ) that when ‖ u ‖ X ≤ R 0 ,

    ‖ S ( t ) u ‖ X ≤ C 0 ( R 0 ) , ( f o r a l l t ∈ [ 0 , + ∞ ) ) ;

  2. there exists a bounded absorbing set B 0 in X , and for any bounded set B ⊂ X , there exists a moment t 0 that

    S ( t ) B ⊂ B 0 ( t ≥ t 0 ) ;

  3. if t > 0 , and S ( t ) is a fully continuous operator, then the semi-group { S ( t ) } t ≥ 0 has a global attractor A in X , and

    A = ω ( B 0 ) = ⋂ τ ≥ 0 ⋃ t ≥ τ S ( t ) B 0 ¯ .

3 Main conclusion

Let ε > 0 be small enough and λ 1 m N 10 − 2 − 4 ε − 2 ε 2 ≥ 0 , λ 1 2 m N 20 − 2 − 4 ε − 2 ε 2 ≥ 0 .

Lemma 3

Assume that assumptions (A1)–(A4) hold, if f j ∈ H ( j = 1 , 2 ) and initial data ( u 0 , u 1 , v 0 , v 1 ) ∈ X 0 × 0 , then for R 0 > 0 , there exist positive constants C ( R 0 ) and t 0 , so that when t ≥ t 0 , ( u , y 1 , v , y 2 ) determined by problems (1)–(3) satisfies

(12) ‖ ( u , y 1 , v , y 2 ) ‖ X 0 × 0 2 = ‖ ∇ m 1 u ‖ 2 + ‖ y 1 ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 + ‖ y 2 ‖ 2 ≤ C ( R 0 ) ,

where y 1 = u t + ε u , y 2 = v t + ε v .

Proof

Multiplying the first equation of (1) by y 1 in H and the second one by y 2 in H , we have

(13) 1 2 d d t ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 + ∫ 0 ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 M ( τ ) d τ + 2 J ( u , v ) + ε M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) − ε ( ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 ) + ε 2 ( ( u , y 1 ) + ( v , y 2 ) ) + N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 y 1 ‖ 2 + N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 y 2 ‖ 2 − ε N 1 ( ‖ ∇ m 1 u ‖ 2 ) ( ∇ m 1 y 1 , ∇ m 1 u ) − ε N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ( ∇ 2 m 2 y 2 , ∇ 2 m 2 v ) + ε ( g 1 ( u , v ) , u ) + ε ( g 2 ( u , v ) , v ) = ( f 1 , y 1 ) + ( f 2 , y 2 ) .

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

(14) − ε ( ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 ) + ε 2 ( ( u , y 1 ) + ( v , y 2 ) ) ≥ − ε − ε 2 2 ( ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 ) − ε 2 2 ( ‖ u ‖ 2 + ‖ v ‖ 2 ) ≥ − ε − ε 2 2 ( ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 ) − ε 2 2 λ 1 − m 1 ‖ ∇ m 1 u ‖ 2 − ε 2 2 λ 1 − 2 m 2 ‖ ∇ 2 m 2 v ‖ 2 ,

(15) N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 y 1 ‖ 2 + N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 y 2 ‖ 2 − ε N 1 ( ‖ ∇ m 1 u ‖ 2 ) ( ∇ m 1 y 1 , ∇ m 1 u ) − ε N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ( ∇ 2 m 2 y 2 , ∇ 2 m 2 v ) ≥ 1 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 y 1 ‖ 2 + 1 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 y 2 ‖ 2 − ε 2 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 u ‖ 2 − ε 2 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 v ‖ 2 ≥ 1 2 λ 1 m 1 N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ y 1 ‖ 2 + 1 2 λ 1 2 m 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ y 2 ‖ 2 − ε 2 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 u ‖ 2 − ε 2 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 v ‖ 2 ,

(16) ( f 1 , y 1 ) + ( f 2 , y 2 ) ≤ ‖ f 1 ‖ ‖ y 1 ‖ + ‖ f 2 ‖ ‖ y 2 ‖ ≤ 1 2 ‖ y 1 ‖ 2 + 1 2 ‖ y 2 ‖ 2 + 1 2 ‖ f 1 ‖ 2 + 1 2 ‖ f 2 ‖ 2 .

Inserting the aforementioned estimates into (13) gives

(17) 1 2 d d t ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 + ∫ 0 ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 M ( τ ) d τ + 2 J ( u , v ) + 1 2 λ 1 m 1 N 1 ( ‖ ∇ m 1 u ‖ 2 ) − 1 2 − ε − ε 2 2 ‖ y 1 ‖ 2 + 1 2 λ 1 2 m 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) − 1 2 − ε − ε 2 2 ‖ y 2 ‖ 2 + ε M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) − ε 2 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) + ε 2 2 λ 1 − m 1 ‖ ∇ m 1 u ‖ 2 − ε 2 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) + ε 2 2 λ 1 − 2 m 2 ‖ ∇ 2 m 2 v ‖ 2 ≤ − ε ( g 1 ( u , v ) , u ) − ε ( g 2 ( u , v ) , v ) + 1 2 ‖ f 1 ‖ 2 + 1 2 ‖ f 2 ‖ 2 .

According to ( A 1 ) , we have

(18) ε M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ≥ ε 4 ∫ 0 ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 M ( τ ) d τ + 3 ε 4 M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ,

and according to ( A 2 ) , we have

(19) − ε ( g 1 ( u , v ) , u ) − ε ( g 2 ( u , v ) , v ) ≤ − ε C 1 J ( u , v ) + ε μ ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) + ε C μ .

Inserting (18) and (19) into (17) gives

(20) d d t ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 + ∫ 0 ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 M ( τ ) d τ + 2 J ( u , v ) + ( λ 1 m 1 N 1 ( ‖ ∇ m 1 u ‖ 2 ) − 1 − 2 ε − ε 2 ) ‖ y 1 ‖ 2 + ( λ 1 2 m 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) − 1 − 2 ε − ε 2 ) ‖ y 2 ‖ 2 + ε 2 ∫ 0 ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 M ( τ ) d τ + 2 ε C 1 J ( u , v ) + 3 ε 2 M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) − 2 ε μ − ε 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) − ε 2 λ 1 − m 1 ‖ ∇ m 1 u ‖ 2 + 3 ε 2 M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) − 2 ε μ − ε 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) − ε 2 λ 1 − 2 m 2 ‖ ∇ 2 m 2 v ‖ 2 ≤ 2 ε C μ + ‖ f 1 ‖ 2 + ‖ f 2 ‖ 2 .

Let H 1 ( t ) = ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 + ∫ 0 ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 M ( τ ) d τ + 2 J ( u , v ) , and σ 1 = min { λ 1 m 1 N 10 − 1 − 2 ε − ε 2 , λ 1 2 m 2 N 20 − 1 − 2 ε − ε 2 , ε 2 , ε C 1 } , we can infer from (20) that

(21) d d t H 1 ( t ) + σ 1 H 1 ( t ) ≤ 2 ε C μ + ‖ f 1 ‖ 2 + ‖ f 2 ‖ 2 .

According to Gronwall’s inequality, we have

(22) H 1 ( t ) ≤ H 1 ( 0 ) e − σ 1 t + 2 ε C μ + ‖ f 1 ‖ 2 + ‖ f 2 ‖ 2 σ 1 ,

and according to ( A 1 ) ( A 2 ) , we have

(23) H 1 ( t ) ≥ ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 + M 0 ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) + 2 J ( u , v ) ≥ ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 + M 0 2 ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) − 2 C μ ′ ≥ C 5 ( ‖ y 1 ‖ 2 + ‖ y 2 ‖ 2 + ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ∣ 2 ) − 2 C μ ′ ,

where μ = M 0 4 , C 5 = min 1 , M 0 2 , then

(24) ‖ ( u , y 1 , v , y 2 ) ‖ X 0 × 0 2 = ‖ ∇ m 1 u ‖ 2 + ‖ y 1 ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 + ‖ y 2 ‖ 2 ≤ ( H 1 ( t ) + 2 C μ ′ ) C 5 ≤ H 1 ( 0 ) e − σ 1 t + 2 C μ ′ C 5 + 2 ε C μ + ‖ f 1 ‖ 2 + ‖ f 2 ‖ 2 σ 1 C 5 ,

i.e.,

(25) lim t → ∞ ¯ ‖ ( u , y 1 , v , y 2 ) ‖ X 0 × 0 2 ≤ 2 C μ ′ C 5 + 2 ε C μ + ‖ f 1 ‖ 2 + ‖ f 2 ‖ 2 σ 1 C 5 = R 0 .

Therefore, there exist positive constants C ( R 0 ) and t 0 that when t ≥ t 0 , we have

(26) ‖ ( u , y 1 , v , y 2 ) ‖ X 0 × 0 2 = ‖ ∇ m 1 u ‖ 2 + ‖ y 1 ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 + ‖ y 2 ‖ 2 ≤ C ( R 0 ) .

Thus, Lemma 3 is proved.□

Lemma 4

Assume that assumptions (A1)–(A4) hold, if f 1 ∈ V k 1 , f 2 ∈ V 2 k 2 , k 1 = 1 , 2 , … , m 1 , k 2 = 1 , 2 , … , m 2 , and initial data ( u 0 , u 1 , v 0 , v 1 ) ∈ X k 1 × k 2 . Then for R k 1 × k 2 > 0 , there exist positive constants C ( R k 1 × k 2 ) and t k 1 × k 2 that when t ≥ t k 1 × k 2 , ( u , y 1 , v , y 2 ) determined by problems (1)–(3) satisfies

(27) ‖ ( u , y 1 , v , y 2 ) ‖ X k 1 × k 2 2 = ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 ≤ C ( R k 1 × k 2 ) ,

where y 1 = u t + ε u , y 2 = v t + ε v .

Proof

Multiplying the first equation of (1) by ( − Δ ) k 1 y 1 , k 1 = 1 , 2 , … , m 1 in H , the second one by ( − Δ ) 2 k 2 y 2 , k 2 = 1 , 2 , … , m 2 in H , and then integrating over Ω , we have

(28) 1 2 d d t [ ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 + M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) ] + ε M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) − ε ( ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 ) + ε 2 ( ( ∇ k 1 u , ∇ k 1 y 1 ) + ( ∇ 2 k 2 v , ∇ 2 k 2 y 2 ) ) + N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 + k 1 y 1 ‖ 2 + N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 + 2 k 2 y 2 ‖ 2 − ε N 1 ( ‖ ∇ m 1 u ‖ 2 ) ( ∇ m 1 + k 1 y 1 , ∇ m 1 + k 1 u ) − ε N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ( ∇ 2 m 2 + 2 k 2 y 2 , ∇ 2 m 2 + 2 k 2 v ) + ( g 1 ( u , v ) , ( − Δ ) k 1 y 1 ) + ( g 2 ( u , v ) , ( − Δ ) 2 k 2 y 2 ) = ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 2 d d t M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) + ( f 1 , ( − Δ ) k 1 y 1 ) + ( f 2 , ( − Δ ) 2 k 2 y 2 ) .

According to Holder’s inequality, Young’s inequality, and Poincare inequality, we have

(29) − ε ( ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 ) + ε 2 ( ( ∇ k 1 u , ∇ k 1 y 1 ) + ( ∇ 2 k 2 v , ∇ 2 k 2 y 2 ) ) ≥ − ε − ε 2 2 ( ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 ) − ε 2 2 ( ‖ ∇ k 1 u ‖ 2 + ‖ ∇ 2 k 2 v ‖ 2 ) ≥ − ε − ε 2 2 ( ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 ) − ε 2 2 λ 1 − m 1 ‖ ∇ m 1 + k 1 u ‖ 2 − ε 2 2 λ 1 − 2 m 2 ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ,

(30) N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 + k 1 y 1 ‖ 2 + N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 + 2 k 2 y 2 ‖ 2 − ε N 1 ( ‖ ∇ m 1 u ‖ 2 ) ( ∇ m 1 + k 1 y 1 , ∇ m 1 + k 1 u ) − ε N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ( ∇ 2 m 2 + 2 k 2 y 2 , ∇ 2 m 2 + 2 k 2 v ) ≥ 1 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 + k 1 y 1 ‖ 2 + 1 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 + 2 k 2 y 2 ‖ 2 − ε 2 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 + k 1 u ‖ 2 − ε 2 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ,

(31) ( g 1 ( u , v ) , ( − Δ ) k 1 y 1 ) + ( g 2 ( u , v ) , ( − Δ ) 2 k 2 y 2 ) ≤ ‖ g 1 ( u , v ) ‖ ‖ ∇ 2 k 2 y 1 ‖ + ‖ g 2 ( u , v ) ‖ ‖ ∇ 4 k 2 y 2 ‖ ≤ N 10 4 ‖ ∇ m 1 + k 1 y 1 ‖ 2 + λ 1 k 1 − m 1 N 10 ‖ g 1 ( u , v ) ‖ 2 + N 20 4 ‖ ∇ 2 m 2 + 2 k 2 y 2 ‖ 2 + λ 1 2 k 2 − 2 m 2 N 20 ‖ g 2 ( u , v ) ‖ 2 ,

(32) ( f 1 , ( − Δ ) k 1 y 1 ) + ( f 2 , ( − Δ ) 2 k 2 y 2 ) ≤ ‖ ∇ k 1 f 1 ‖ ‖ ∇ k 1 y 1 ‖ + ‖ ∇ 2 k 2 f 2 ‖ ‖ ∇ 2 k 2 y 2 ‖ ≤ 1 2 ‖ ∇ k 1 y 1 ‖ 2 + 1 2 ‖ ∇ 2 k 2 y 2 ‖ 2 + 1 2 ‖ ∇ k 1 f 1 ‖ 2 + 1 2 ‖ ∇ 2 k 2 f 2 ‖ 2 ,

and according to ( A 3 ) , we have

(33) ‖ g 1 ( u , v ) ‖ 2 = ∫ Ω ∣ g 1 ( u , v ) ∣ 2 d x ≤ ∫ Ω ∣ C 2 ( 1 + ∣ u ∣ p 1 + ∣ v ∣ q 1 ) ∣ 2 d x ≤ C 6 ∫ Ω ( 1 + ∣ u ∣ 2 p 1 + ∣ v ∣ 2 q 1 ) d x ≤ C 7 ( 1 + ‖ u ‖ 2 p 1 2 p 1 + ‖ v ‖ 2 q 1 2 q 1 ) , ‖ g 2 ( u , v ) ‖ 2 ≤ C 8 ( 1 + ‖ u ‖ 2 p 2 2 p 2 + ‖ v ‖ 2 q 2 2 q 2 ) .

Furthermore, on the basis of the Gagliardo-Nirenberg inequality, we can conclude that

‖ u ‖ 2 p j 2 p j ≤ C 9 j ‖ ∇ m 1 u ‖ n ( p j − 1 ) m 1 ‖ u ‖ 2 m 1 p j − n ( p j − 1 ) m 1 , ‖ v ‖ 2 q j 2 q j ≤ C 10 j ‖ ∇ 2 m 2 v ‖ n ( q j − 1 ) 2 m 2 ‖ v ‖ 4 m 2 q j − n ( q j − 1 ) 2 m 2 .

Thus, we have

(34) ‖ g 1 ( u , v ) ‖ 2 + ‖ g 2 ( u , v ) ‖ 2 ≤ C ( R 0 ) .

By inserting (29)–(32) and (34) into (28), we obtain

(35) 1 2 d d t [ ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 + M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) ] + ( 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) − N 10 ) λ 1 m 1 − 2 − 4 ε − 2 ε 2 4 ‖ ∇ k 1 y 1 ‖ 2 + ( 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) − N 20 ) λ 1 2 m 2 − 2 − 4 ε − 2 ε 2 4 × ‖ ∇ 2 k 2 y 2 ‖ 2 + ( ε M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) − ε 2 2 N 1 ( ‖ ∇ m 1 u ‖ 2 ) − ε 2 2 λ 1 − m 1 ) ‖ ∇ m 1 + k 1 u ‖ 2 + ( ε M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) − ε 2 2 N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) − ε 2 2 λ 1 − 2 m 2 ) ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ≤ ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 2 d d t M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) + 1 2 ‖ ∇ k 1 f 1 ‖ 2 + 1 2 ‖ ∇ 2 k 2 f 2 ‖ 2 + C ( R 0 , λ 1 ) ≤ ( ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) M ′ ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) × ( ( ∇ m 1 u , ∇ m 1 u t ) + ( ∇ 2 m 2 v , ∇ 2 m 2 v t ) ) + 1 2 ‖ ∇ k 1 f 1 ‖ 2 + 1 2 ‖ ∇ 2 k 2 f 2 ‖ 2 + C ( R 0 , λ 1 ) ≤ C 8 ( ‖ ∇ m 1 u t ‖ + ‖ ∇ 2 m 2 v t ‖ ) ( ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) + 1 2 ‖ ∇ k 1 f 1 ‖ 2 + 1 2 ‖ ∇ 2 k 2 f 2 ‖ 2 + C ( R 0 , λ 1 ) .

Let H 2 ( t ) = ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 + M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) , and σ 2 = min { λ 1 m 1 N 10 − 2 − 4 ε − 2 ε 2 2 , λ 1 2 m 2 N 20 − 2 − 4 ε − 2 ε 2 2 , ε 2 } , we have

(36) d d t H 2 ( t ) + σ 2 H 2 ( t ) ≤ C 9 ( ‖ ∇ m 1 u t ‖ + ‖ ∇ 2 m 2 v t ‖ ) H 2 ( t ) + ‖ ∇ k 1 f 1 ‖ 2 + ‖ ∇ 2 k 2 f 2 ‖ 2 + C ( R 0 , λ 1 ) .

By taking the scalar product in H of (1) with u t , v t , we have

(37) 1 2 d d t ‖ u t ‖ 2 + ‖ v t ‖ 2 + ∫ 0 ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 M ( τ ) d τ + 2 J ( u , v ) − 2 ( f 1 , u ) − 2 ( f 2 , v ) + N 1 ( ‖ ∇ m 1 u ‖ 2 ) ‖ ∇ m 1 u t ‖ 2 + N 2 ( ‖ ∇ 2 m 2 v ‖ 2 ) ‖ ∇ 2 m 2 v t ‖ 2 = 0 ,

and integrating (37) in d t on ( 0 , t ) derives

(38) ∫ 0 t ( ‖ ∇ m 1 u t ( τ ) ‖ 2 + ‖ ∇ 2 m 2 v t ( τ ) ‖ 2 ) d τ ≤ 1 min { N 10 , N 20 } ∫ 0 t ( N 1 ( ‖ ∇ m 1 u ( τ ) ‖ 2 ) ‖ ∇ m 1 u t ( τ ) ‖ 2 + N 2 ( ‖ ∇ 2 m 2 v ( τ ) ‖ 2 ) ‖ ∇ 2 m 2 v t ( τ ) ‖ 2 ) d τ ≤ 1 min { N 10 , N 20 } ‖ u 1 ‖ 2 + ‖ v 1 ‖ 2 + ∫ 0 ‖ ∇ m 1 u 0 ‖ 2 + ‖ ∇ 2 m 2 v 0 ‖ 2 M ( τ ) d τ + 2 J ( u 0 , v 0 ) − 2 ( f 1 , u 0 ) − 2 ( f 2 , v 0 ) ≤ C 10 ,

then, we have

(39) C 9 ∫ s t ( ( ‖ ∇ m 1 u t ( τ ) ‖ + ‖ ∇ 2 m 2 v t ( τ ) ‖ ) ) d τ ≤ σ 2 2 ( t − s ) + a ,

for t > s ≥ 0 and some a > 0 . Together with (36), (39), and Lemma 1, we can obtain that

(40) H 2 ( t ) ≤ C 11 H 2 ( 0 ) e − σ 2 2 t + C 12 .

According to ( A 1 ) , we have

(41) H 2 ( t ) ≥ ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 + M ( ‖ ∇ m 1 u ‖ 2 + ‖ ∇ 2 m 2 v ‖ 2 ) ( ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) ≥ C 13 ( ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 + ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ) ,

then,

(42) ‖ ( u , y 1 , v , y 2 ) ‖ X k 1 × k 2 2 = ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 + ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ≤ C 11 H 2 ( 0 ) e − σ 2 2 t + C 12 C 13 ,

i.e.,

(43) lim t → ∞ ¯ ‖ ( u , y 1 , v , y 2 ) ‖ X k 1 × k 2 2 ≤ R k 1 × k 2 .

Therefore, there exist positive constants C ( R k 1 × k 2 ) and t k 1 × k 2 that when t ≥ t k 1 × k 2 , ( u , y 1 , v , y 2 ) satisfies

(44) ‖ ( u , y 1 , v , y 2 ) ‖ X k 1 × k 2 2 = ‖ ∇ k 1 y 1 ‖ 2 + ‖ ∇ 2 k 2 y 2 ‖ 2 + ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 ≤ C ( R k 1 × k 2 ) , k 1 = 1 , 2 , … , m 1 , k 2 = 1 , 2 , … , m 2 .

Thus, Lemma 4 is proved.□

Theorem 1

Assume that assumptions (A1)–(A4) hold, if f 1 ∈ V k 1 , f 2 ∈ V 2 k 2 and initial data ( u 0 , u 1 , v 0 , v 1 ) ∈ X k 1 × k 2 , k 1 = 0 , 1 , 2 , … , m 1 , k 2 = 0 , 1 , 2 , … , m 2 , then problems (1)–(3) admit a unique solution ( u , v ) satisfying

u ∈ L ∞ ( 0 , ∞ ; V m 1 + k 1 ) ; u t ∈ L ∞ ( 0 , ∞ ; H ) ∩ L 2 ( 0 , T ; V k 1 ) ; v ∈ L ∞ ( 0 , ∞ ; V 2 m 2 + 2 k 2 ) ; v t ∈ L ∞ ( 0 , ∞ ; H ) ∩ L 2 ( 0 , T ; V 2 k 2 ) .

Proof

According to [15] and the Faedo-Galerkin method, (1)–(3) have global solutions combining with Lemmas 3 and 4.

Let u 1 , v 1 and u 2 , v 2 be two solutions of problems (1)–(3) corresponding to the same initial data, respectively, w = u 1 − u 2 , z = v 1 − v 2 . Then, ( w , z ) solves

(45) w t t + 1 2 σ 12 ( t ) ( − Δ ) m 1 w t + 1 2 Φ 12 ( t ) ( − Δ ) m 1 w + G 1 ( u 1 , u 2 , v 1 , v 2 ; t ) = 0 , z t t + 1 2 σ 34 ( t ) ( − Δ ) 2 m 2 z t + 1 2 Φ 12 ( t ) ( − Δ ) 2 m 2 z + G 2 ( u 1 , u 2 , v 1 , v 2 ; t ) = 0 ,

where σ 12 = σ 1 ( t ) + σ 2 ( t ) , Φ 12 ( t ) = Φ 1 ( t ) + Φ 2 ( t ) , σ i ( t ) = N 1 ( ‖ ∇ m 1 u i ‖ 2 ) , Φ i ( t ) = M ( ‖ ∇ m 1 u i ‖ 2 + ‖ ∇ 2 m 2 v i ‖ 2 ) , i = 1 , 2 , σ 34 = σ 3 ( t ) + σ 4 ( t ) , σ j ( t ) = N 2 ( ‖ ∇ 2 m 2 v j ‖ 2 ) , j = 3 , 4 , G 1 ( u 1 , u 2 , v 1 , v 2 ; t ) = 1 2 { [ σ 1 ( t ) − σ 2 ( t ) ] ( − Δ ) m 1 ( u t 1 + u t 2 ) + [ Φ 1 ( t ) − Φ 2 ( t ) ] ( − Δ ) m 1 ( u 1 + u 2 ) } + g 1 ( u 1 , v 1 ) − g 1 ( u 2 , v 2 ) , G 2 ( u 1 , u 2 , v 1 , v 2 ; t ) = 1 2 { [ σ 3 ( t ) − σ 4 ( t ) ] ( − Δ ) 2 m 2 ( v t 1 + v t 2 ) + [ Φ 1 ( t ) − Φ 2 ( t ) ] ( − Δ ) 2 m 2 ( v 1 + v 2 ) } + g 2 ( u 1 , v 1 ) − g 2 ( u 2 , v 2 ) .

According to Lemma 3, σ 12 ′ ≤ C ( R 0 ) ( ‖ ∇ m 1 u t 1 ‖ + ‖ ∇ m 1 u t 2 ‖ ) , σ 34 ′ ≤ C ( R 0 ) ( ‖ ∇ 2 m 2 v t 1 ‖ + ‖ ∇ 2 m 2 v t 2 ‖ ) .

By taking the scalar product in H of (45) with w t , z t , we can obtain that

(46) 1 2 d d t ‖ w t ‖ 2 + ‖ z t ‖ 2 + 1 4 Φ 0 ⋅ ( ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) + 1 2 σ 12 ( t ) ‖ ∇ m 1 w t ‖ 2 + 1 2 σ 34 ( t ) ‖ ∇ 2 m 2 z t ‖ 2 + ( G 1 ( u 1 , u 2 , v 1 , v 2 ; t ) , w t ) + ( G 2 ( u 1 , u 2 , v 1 , v 2 ; t ) , z t ) = 0 .

According to Lemma 3 and ( A 1 ) , M 0 ≤ M ≤ C ( R 0 , H 1 ( 0 ) ) ≡ M 1 . When d d t ( ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) ≥ 0 , Φ 0 = 2 M 0 ; otherwise Φ 0 = 2 M 1 .

Let ( G 1 ( u 1 , u 2 , v 1 , v 2 ; t ) , w t ) = G 11 + G 12 + G 13 , ( G 2 ( u 1 , u 2 , v 1 , v 2 ; t ) , z t ) = G 21 + G 22 + G 23 , we have

(47) G 11 = 1 2 ( σ 1 ( t ) − σ 2 ( t ) ) ( ∇ m 1 ( u t 1 + u t 2 ) , ∇ m 1 w t ) ≤ C ( R 0 ) ( ‖ ∇ m 1 u t 1 ‖ + ‖ ∇ m 1 u t 2 ‖ ) ‖ ∇ m 1 w ‖ ‖ ∇ m 1 w t ‖ ≤ σ 120 8 ‖ ∇ m 1 w t ‖ 2 + 2 C ( R 0 ) σ 120 ( ‖ ∇ m 1 u t 1 ‖ 2 + ‖ ∇ m 1 u t 2 ‖ 2 ) ‖ ∇ m 1 w ‖ 2 ,

(48) G 12 = 1 2 ( Φ 1 ( t ) − Φ 2 ( t ) ) ( ∇ m 1 ( u 1 + u 2 ) , ∇ m 1 w t ) ≤ C ( R 0 ) ( ‖ ∇ m 1 w ‖ + ‖ ∇ 2 m 2 z ‖ ) ‖ ∇ m 1 w t ‖ ≤ σ 120 8 ‖ ∇ m 1 w t ‖ 2 + 2 C ( R 0 ) σ 120 ( ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) ,

(49) G 13 = ( g 1 ( u 1 , v 1 ) − g 1 ( u 2 , v 2 ) , w t ) ≤ C ( R 0 ) ( ‖ w t ‖ 2 + ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) ,

(50) G 21 = 1 2 ( σ 3 ( t ) − σ 4 ( t ) ) ( ∇ 2 m 2 ( v t 1 + v t 2 ) , ∇ 2 m 2 z t ) ≤ C ( R 0 ) ( ‖ ∇ 2 m 2 v t 1 ‖ + ‖ ∇ 2 m 2 v t 2 ‖ ) ‖ ∇ 2 m 2 z ‖ ‖ ∇ 2 m 2 z t ‖ ≤ σ 340 8 ‖ ∇ 2 m 2 z t ‖ 2 + 2 C ( R 0 ) σ 340 ( ‖ ∇ 2 m 2 v t 1 ‖ 2 + ‖ ∇ 2 m 2 v t 2 ‖ 2 ) ‖ ∇ 2 m 2 z ‖ 2 ,

(51) G 22 = 1 2 ( Φ 1 ( t ) − Φ 2 ( t ) ) ( ∇ 2 m 2 ( v 1 + v 2 ) , ∇ 2 m 2 z t ) ≤ C ( R 0 ) ( ‖ ∇ m 1 w ‖ + ‖ ∇ 2 m 2 z ‖ ) ‖ ∇ 2 m 2 z t ‖ ≤ σ 340 8 ‖ ∇ 2 m 2 z t ‖ 2 + 2 C ( R 0 ) σ 340 ( ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) ,

(52) G 23 = ( g 2 ( u 1 , v 1 ) − g 2 ( u 2 , v 2 ) , z t ) ≤ C ( R 0 ) ( ‖ z t ‖ 2 + ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) ,

where σ 120 = 2 N 10 , σ 340 = 2 N 20 .

By inserting (46)–(52) into (45), we have

(53) 1 2 d d t ‖ w t ‖ 2 + ‖ z t ‖ 2 + 1 4 Φ 0 ⋅ ( ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) ≤ C 14 ( 1 + ‖ ∇ 2 m 2 v t 1 ‖ 2 + ‖ ∇ 2 m 2 v t 2 ‖ 2 ) ‖ w t ‖ 2 + ‖ z t ‖ 2 + 1 4 Φ 0 ⋅ ( ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) .

Solving this differential inequality yields

(54) ‖ w t ‖ 2 + ‖ z t ‖ 2 + 1 4 Φ 0 ⋅ ( ‖ ∇ m 1 w ‖ 2 + ‖ ∇ 2 m 2 z ‖ 2 ) ≤ ‖ w 1 ‖ 2 + ‖ z 1 ‖ 2 + 1 4 Φ 0 ⋅ ( ‖ ∇ m 1 w 0 ‖ 2 + ‖ ∇ 2 m 2 z 0 ‖ 2 ) exp ∫ 0 t C 14 ( 1 + ‖ ∇ 2 m 2 v t 1 ‖ 2 + ‖ ∇ 2 m 2 v t 2 ‖ 2 ) d s .

Thus, the uniqueness of the solution is proved.□

Therefore, problems (1)–(3) possess a unique solution u , v . Theorem 1 is proved.

According to Theorem 1, we define a nonlinear operator { S ( t ) } t ≥ 0 on space X 0 × 0 : S ( t ) ( u 0 , u 1 , v 0 , v 1 ) = ( u , u t , v , v t ) , for all t ≥ 0 . Theorem 1 shows that { S ( t ) } t ≥ 0 compose a continuous semi-group in X 0 × 0 . Before proving the family of global attractors, we first give their definition.

Definition 1

Let X 0 be a Banach space, and { S ( t ) } t ≥ 0 be a continuous operator semi-group, if there exist compact sets A k 1 × k 2 that satisfies

  1. Invariance: all A k 1 × k 2 are invariant sets under the action of the semi-group { S ( t ) } t ≥ 0 ,

    S ( t ) A k 1 × k 2 = A k 1 × k 2 ; for all t ≥ 0 .

  2. Attractiveness: all A k 1 × k 2 attract all bounded sets in X 0 , i.e., for any bounded B ⊂ X 0 ,

    dist ( S ( t ) B , A k 1 × k 2 ) = sup x ∈ B inf y ∈ A k 1 × k 2 ‖ S ( t ) x − y ‖ X 0 → 0 , t → ∞ .

In particular, when t → ∞ , all trajectories S ( t ) u 0 from u 0 converge to A k 1 × k 2 , i.e.,

dist ( S ( t ) u 0 , A k 1 × k 2 ) → 0 , t → ∞ .

Then, compact sets A k are all global attractors of the semi-group { S ( t ) } t ≥ 0 . Let A = { A k 1 × k 2 ⊂ X 0 : k 1 = 1 , 2 , … , m 1 , k 2 = 1 , 2 , … , m 2 } be a family of subsets in X 0 . Finally, the family A is the family of global attractors in X 0 .

Theorem 2

Assume that assumptions (A1)–(A4) hold, if f 1 ∈ V m 1 , f 2 ∈ V 2 m 2 and initial data ( u 0 , u 1 , v 0 , v 1 ) ∈ X m 1 × m 2 , then, problems (1)–(3) have a family of global attractors A in X 0 × 0

A = { A k 1 × k 2 } , A k 1 × k 2 = ω ( B k 1 × k 2 , 0 ) = ⋂ τ ≥ 0 ⋃ t ≥ τ S ( t ) B k 1 × k 2 , 0 ¯ , k 1 = 1 , 2 , … , m 1 , k 2 = 1 , 2 , … , m 2 ,

where B k 1 × k 2 , 0 = { ( u , u t , v , v t ) ∈ X k 1 × k 2 : ‖ ( u , u t , v , v t ) ‖ X k 1 × k 2 2 = ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ k 1 u t ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 + ‖ ∇ 2 k 2 v t ‖ 2 ≤ C ( R 0 ) + C ( R k 1 × k 2 ) } are bounded absorbing sets in X 0 × 0 , B k 1 × k 2 , 0 are compact in X 0 × 0 , A k 1 × k 2 ⊂ X 0 × 0 .

  1. S ( t ) A k 1 × k 2 = A k 1 × k 2 , (for all t ≥ 0 ),

  2. A k 1 × k 2 attract all bounded sets in X 0 × 0 , i.e., for all B k 1 × k 2 ⊂ X 0 × 0 are bounded sets in X 0 × 0 , and

    dist ( S ( t ) B k 1 × k 2 , A k 1 × k 2 ) = sup x ∈ B k 1 × k 2 inf y ∈ A k 1 × k 2 ‖ S ( t ) x − y ‖ X 0 × 0 → 0 ( t → ∞ ) ,

    where { S ( t ) } t ≥ 0 is the solution semi-group generated by problems (1)–(3).

Proof

According to Lemma 3, for all R 0 > 0 , ‖ ( u 0 , u 1 , v 0 , v 1 ) ‖ X 0 × 0 ≤ R 0 . Thus,

‖ S ( t ) ( u 0 , u 1 , v 0 , v 1 ) ‖ X 0 × 0 2 = ‖ u ‖ V m 1 2 + ‖ u t ‖ V 0 2 + ‖ v ‖ V 2 m 2 2 + ‖ v t ‖ V 0 2 ≤ C ( R 0 )

show that { S ( t ) } t ≥ 0 are uniformly bounded in X 0 × 0 ; further, B k 1 × k 2 , 0 = { ( u , u t , v , v t ) ∈ X k 1 × k 2 : ‖ ( u , u t , v , v t ) ‖ X k 1 × k 2 2 = ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ k 1 u t ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 + ‖ ∇ 2 k 2 v t ‖ 2 ≤ C ( R 0 ) + C ( R k 1 × k 2 ) } are bounded absorbing sets of the semi-group { S ( t ) } t ≥ 0 in X 0 × 0 ; because X k 1 × k 2 ↪ X 0 × 0 are compactly embedding, i.e., the bounded sets in X k 1 × k 2 are compact sets in X 0 × 0 , the solution semi-group { S ( t ) } t ≥ 0 is a fully continuous operator.

To sum up, we obtained the family of global attractors A = { A k 1 × k 2 } of the solution semi-group { S ( t ) } t ≥ 0 in X 0 × 0 , and

A k 1 × k 2 = ω ( B k 1 × k 2 , 0 ) = ⋂ τ ≥ 0 ⋃ t ≥ τ S ( t ) B k 1 × k 2 , 0 ¯ , A k 1 × k 2 ⊂ X 0 × 0 , k 1 = 1 , 2 , … , m 1 , k 2 = 1 , 2 , … , m 2 .

Theorem 2 is proved.□

Note 1. Lemma 4 and Theorem 2 show that the bounded absorbing sets B k 1 × k 2 , 0 = { ( u , u t , v , v t ) ∈ X k 1 × k 2 : ‖ ( u , u t , v , v t ) ‖ X k 1 × k 2 2 = ‖ ∇ m 1 + k 1 u ‖ 2 + ‖ ∇ k 1 u t ‖ 2 + ‖ ∇ 2 m 2 + 2 k 2 v ‖ 2 + ‖ ∇ 2 k 2 v t ‖ 2 ≤ C ( R 0 ) + C ( R k 1 × k 2 ) } are compact bounded absorbing sets in X 0 × 0 . Therefore, based on condition 3 in Lemma 2, the operator semi-group { S ( t ) } t ≥ 0 only needs to be a continuous operator. According to Theorem 1, the semi-group { S ( t ) } t ≥ 0 is already a continuous semi-group. Thus, the family of global attractors A = { A k 1 × k 2 } of problems (1)–(3) in X 0 × 0 can also be obtained.

4 Conclusion

In this article, we studied a class of higher-order coupled Kirchhoff systems with nonlinear strong damping. For the first time, we systematically defined the family of global attractors and proved the existence of the family of global attractors of problems (1)–(3). The results enriched the related conclusions of higher-order coupled Kirchhoff models and laid a theoretical foundation for future practical applications.

Acknowledgments

The authors would like to thank the anonymous referees for their comments and suggestions, which helped greatly improve the article.

  1. Funding information: This work was partially supported by the basic science (Natural Science) research project of colleges and universities in Jiangsu Province (21KJB110013) and the fundamental research fund of Yunnan Education Department (2020J0908).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.

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Received: 2022-10-10
Revised: 2023-01-01
Accepted: 2023-01-24
Published Online: 2023-03-28

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

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  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
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https://doi.org/10.1515/dema-2022-0197
Keywords for this article
higher-order coupled Kirchhoff system; nonlinear strong damping; global attractor family
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Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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