Long-Term Dynamic Behavior of a Higher-Order Coupled Kirchhoff Model with Nonlinear Strong Damping

In this paper, we study the long-time dynamics problem of a class of higher-order Kirchho coupled systems with nonlinear strong damping. e existence and uniqueness of the solutions of these equations in dierent spaces are proved by prior estimation and Faedo–Galerkin method; secondly, the family of global attractors of these problems is proved by using the compactness theorem. e results of the Kirchho coupled group are promoted through research.


Introduction
In this paper, we mainly consider the dynamic behavior of the higher-order Kirchho -type coupled equations on the bounded smooth domain Ω ⊂ R N : with boundary conditions: and initial conditions: is is a kind of important generalized quasilinear wave equations of higher order. e proposed equation in this paper originated from Kirchho 's vibration problem of stretchable strings in 1883: where 0 < x < L, t ≥ 0, u � u(x, t) is the lateral displacement in space coordinate x and time coordinate t, E represents 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. e research studies on the long-time behavior of various forms of Kirchhoff-type equations have attracted the attention of many scholars in recent decades and have also achieved rich results [1][2][3][4][5][6][7][8]. Igor Chueshov [1] studied the well-posedness and long-time dynamical behavior of the following Kirchhoff equation with a nonlinear strong damping term: Guoguang Lin, Penghui Lv, and Ruijin Lou [2] studied the global dynamics of the following generalized nonlinear Kirchhoff-Boussinesq equations with damping terms: is paper proved that the semigroup conformed to the squeezing property and then obtained the existence of the exponential attractor of the system; then, it used the spectral interval theory to prove that the system had an inertial manifold.
Marina Ghisi and Massimo Gobbino [3] studied the global existence and local existence of solutions to the following Kirchhoff model with strong damping: Mitsuhiro Nakao [4] proved the initial-boundary value problem of the quasilinear Kirchhoff-type wave equation with standard dissipation u t : With the deepening of research, scholars began to turn their research directions to the dynamics of the higher-order Kirchhoff equations. Ye Yaojun and Tao Xiangxing [9] studied the initial-boundary value problem of the following kind of higher-order Kirchhoff-type equation with nonlinear dissipation term: Lin Guoguang and Zhu Changqing [10] studied the initial and boundary value problems of the following nonlinear nonlocal higher-order Kirchhoff-type equations: e paper obtained the existence and uniqueness of the solution and obtained the existence of the family of global attractors of the problem through the compact method and obtained the finite Hausdorff and Fractal dimensions.
System coupling originates from physics and is a metric used to refer to the mutual dependence of two entities on each other. Coupled systems refer to the system coupling occurring when some conditions or parameters are appropriate, and the potential energy of the system can make different systems realize the combination of structure and function and then produce new functions. e Kirchhoff models are mathematical equations derived from a physical background, and it will be natural to consider their coupled systems. Later, scholars gradually consider the dynamics of the Kirchhoff equations under the coupled effect. For example, Wang Yu and Zhang Jianwen [11] studied the longterm dynamics problem of a class of coupled beam equations with strong damping under nonlinear boundary conditions. Guoguang Lin and Ming Zhang [12] have studied the initialboundary value problem of the following Kirchhoff coupling group with strong damping and source term: In this paper, the finite Hausdorff dimension of the global attractor is obtained.
In recent years, Guoguang Lin et al. [13][14][15] focused on the dynamics of a class of higher-order Kirchhoff coupled equations and obtained a series of ideal results. More conclusions about higher-order Kirchhoff-type systems can also be found in [16][17][18][19].
For the higher-order Kirchhoff coupled problems, there are few articles at present, and the problem of higher-order beam-plate coupled with nonlinear strong damping has not been studied. e main difficulty lies in the estimation and processing of the harmonic term and the nonlinear damping term, and when proving the uniqueness, the nonlinear damping brings some difficulties. In this regard, this paper overcomes these difficulties and obtains the global solution of the problem and the family of global attractors.

Preparatory Knowledge
In this paper, we use ‖ · ‖ and (·, ·) to represent the norm and inner product in H � L 2 (Ω), respectively. In order to get a more ideal conclusion, given the following assumptions: (iv) (A4).N j (s j ) ≥ N j0 , N j0 (j � 1, 2) are positive constants, and there exists ρ > 0, so that Next, the research phase space of this paper is given as follows: and the norms of the corresponding spaces are as follows: Meanwhile, there exists the general form of Poincare's inequality: 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 paper, C i is a constant, and C(·) is a constant that depends on the parameters in parentheses.
Lemma 1 (see [20]). Let y: R + ⟶ R + be an absolutely continuous positive function on [0, +∞), which satisfies for some δ > 0 the differential inequality where Lemma 2 (see [10]). Let X be a Banach space, and the continuous operator semigroup S(t) { } t≥0 satisfies the following: (2) ere exists a bounded absorbing set B 0 in X, and for any bounded set B ⊂ X, there exists a moment t 0 such that