The asymptotic behaviors of solutions for higher - order ( m 1 , m 2 )- coupled Kirchho ﬀ models with nonlinear strong damping

: The Kirchho ﬀ 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 Kirchho ﬀ systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in di ﬀ erent 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 de ﬁ nition and proof process of the family of global attractors and enrich the related conclusions of higher - order coupled Kirchho ﬀ models. The conclusions lay a theoretical foundation for future practical applications.


Introduction
In this study, we consider the dynamic behavior of the following higher-order coupled Kirchhoff models in a bounded smooth domain R Ω n ⊂ : under the following boundary conditions: and the following initial conditions: , where Δ is the Laplace operator, N N , 1 2 and M M , 1 2 are scalar functions specified later, g g , 1 2 are the given source terms, and f f , 1 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: where x L 0 < < , 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][2][3][4][5][6][7][8][9][10][11][12][13].
Chueshov [1] studied the well-posedness and long-time dynamical behavior of the following Kirchhoff equation with a nonlinear strong damping term: Lin et al. [2] studied the global dynamics of the following generalized nonlinear Kirchhoff-Boussinesq equations with a strong damping: 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: Nakao [4] proved the initial-boundary value problem of the quasi-linear Kirchhoff-type wave equation with standard dissipation u t : 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: Lin and Zhu [15] studied the initial and boundary value problems of the following nonlinear nonlocal higher-order Kirchhoff-type equations: 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: The finite Hausdorff dimension of the global attractor can be obtained in [17].
In recent years, Lin et al. [18][19][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.

Preparatory knowledge
In this article, ‖⋅‖ and , ( ) ⋅ ⋅ denote the norm and the inner product in 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, , Δ , Δ ( ) ‖⋅‖ = ‖ − ⋅‖.The main goal is to study the well-posedness and long-time dynamics of problems (1) to (3) under the following set of hypotheses: are positive constants, and ρ 0 > .Thus, M s s ρN s Then, the research phase space of this study is obtained: and the norms of the corresponding spaces are as follows: Meanwhile, the general form of the Poincare inequality is as follows: , 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.
be an absolutely continuous positive function on 0, [ ) +∞ , which satisfies the following differential inequality for some δ 0 > : where K 0 ≥ , and a 0 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 R 0 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 (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 ( ) and t 0 , so that when t t 0 ≥ , u y v y , , , ) determined by problems (1)-(3) satisfies where y u εu y v εv , Proof.Multiplying the first equation of (1) by y 1 in H and the second one by y 2 in H , we have By Holder's inequality, Young's inequality, and Poincare inequality, we have , Inserting the aforementioned estimates into (13) gives .
According to A 1 ( ), we have and according to A 2 ( ), we have , , , .
The asymptotic behaviors for higher-order (m 1 ,m 2 )-coupled Kirchhoff models  5 Inserting (18) and ( 19) into (17) gives , and According to Gronwall's inequality, we have and according to A A ( )( ), we have where μ C , min 1, i.e., Therefore, there exist positive constants C R 0 ( ) and t 0 that when t t 0 ≥ , we have Thus, Lemma 3 is proved.
= … , and initial data u u v v X , , , × that when t t k k ) determined by problems (1)-( 3) satisfies where y u εu y v εv , Proof.Multiplying the first equation of (1) by = … in H , and then integrating over Ω, we have According to Holder's inequality, Young's inequality, and Poincare inequality, we have , , ) ) and according to A 3 ( ), we have p q p q p p q q p p q q The asymptotic behaviors for higher-order (m 1 ,m 2 )-coupled Kirchhoff models  7 Furthermore, on the basis of the Gagliardo-Nirenberg inequality, we can conclude that Thus, we have By inserting (29)-( 32) and (34) into (28), we obtain By taking the scalar product in H of (1) with u v , t t , we have and integrating (37 for t s 0 > ≥ and some a 0 > .Together with (36), (39), and Lemma i.e., Therefore, there exist positive constants C R k k Thus, Lemma 4 is proved.