Abstract
In this paper, we study the long-time dynamics problem of a class of higher-order Kirchhoff coupled systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different 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. The results of the Kirchhoff coupled group are promoted through research.
1. Introduction
In this paper, we mainly consider the dynamic behavior of the higher-order Kirchhoff-type coupled equations on the bounded smooth domain :with boundary conditions:and initial conditions:
This is a kind of important generalized quasilinear wave equations of higher order. The proposed equation in this paper originated from Kirchhoff’s vibration problem of stretchable strings in 1883:where is the lateral displacement in space coordinate and time coordinate , represents Young’s modulus, represents the mass density, represents the cross-sectional area, represents the length, and represents the axial tension of the accident. The 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 [1–8]. 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:
This 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 :
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:
The 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. The 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 long-term 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 initial-boundary 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. [13–15] 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 [16–19].
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. The 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.
2. Preparatory Knowledge
In this paper, we use and to represent the norm and inner product in , respectively. In order to get a more ideal conclusion, given the following assumptions:(i) is a continuous function on the interval , , and , .(ii). For any , let , where , then for any , there exists , such that , .(iii), and ; ; . Among them, when , ; when , .(iv), are positive constants, and there exists , 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: , where is the first eigenvalue of with a homogeneous Dirichlet boundary on . In this paper, is a constant, and is a constant that depends on the parameters in parentheses.
Lemma 1 (see [20]). Let be an absolutely continuous positive function on , which satisfies for some the differential inequalitywhere ; for , there exists , such that , then
Lemma 2 (see [10]). Let be a Banach space, and the continuous operator semigroup satisfies the following:(1)The semigroup is uniformly bounded in , that is, ; there exists a positive constant such that when ,(2)There exists a bounded absorbing set in , and for any bounded set , there exists a moment such that(3)For is a fully continuous operator.Then, the semigroup has a global attractor in , and
3. Main Conclusion
Let be small enough and .
Lemma 3. Assume that assumptions hold if and initial data ; then, for , there exist positive constants and , so that when , determined by problems (1)–(3) satisfieswhere .
Proof. Multiplying the first equation of 1 by in and the second one by in , we infer thatUsing Holder’s inequality, Young’s inequality, Poincare’s inequality, and so on to process the items in (20),Substituting (21)–(23) into (20), we can obtainBy , we getand according to , we getCombining (25) and (26), (24) becomesLet and , we infer from (27) thatUsing Gronwall’s inequality, we getAccording to , taking , we havewhere .
Then,and thusTherefore, there exist positive constants and , such that when , we getThe proof of Lemma 3 is completed.
Lemma 4. Assume that assumptions hold, if and initial data ; then, for , there exist positive constants and , so that when , determined by problems (1)–(3) satisfieswhere .
Proof. Multiplying the first equation of 1 by in and the second one by in , we infer thatUsing Holder’s inequality, Young’s inequality, Poincare’s inequality, and so on to process the items in (35),According to , we getand . Furthermore, using the Gagliardo–Nirenberg inequality, we concludethenCombining (36)–(38) and (39), (28) becomesLet and , we haveTaking the scalar product in of (1) with , we obtainand integrating (42) in on , we getand then we haveCombining (41) and (44), according to Lemma 1, we knowBy , we havethenthat is,Therefore, there exist positive constants and , such that when , satisfiesThe proof of Lemma 4 is completed.
Theorem 1. Assume that assumptions hold if and initial data , then problems (1)– (3) have a unique solution satisfying
Proof. According to [10] and the Faedo–Galerkin method, it is easy to obtain that (1)– (3) have global solutions combining with Lemma 3 and Lemma 4.
The uniqueness of the solution is as follows:
Let and be two solutions of problems (1)–(3) with the same initial value. Write , then satisfieswhereFrom Lemma 3, it can be known that and .
Taking the scalar product in of (51) with , we can obtainFrom Lemma 3 and , we know ; when , we write ; otherwise, .
Let , we havewhere .
Combining (53)–(59), (52) can be changed:thenThe uniqueness of the solution is proved.
The proof of Theorem 1 is completed.
According to Theorem 1, we define a nonlinear operator on space . It can be known that is a nonlinear semigroup defined on .
Theorem 2. Assume that assumptions (A1)–(A4) hold, if and initial data , then problems (1)–(3) have a family of global attractors in :where . Obviously, B0k are bounded absorbing sets in X0 and compact in X0.(1),(2) attracts all bounded sets in , that is, are bounded sets in , and is the solution semigroup generated by problems (1)–(3).
Proof. From Lemma 3, , , such thatwhich shows that are uniformly bounded in .
Further, are bounded absorbing sets of the semigroup in because are compactly embedding, that is, the bounded sets in are compact sets in , so the solution semigroup is a fully continuous operator.
To sum up, we get the family of global attractors of the solution semigroup in , andThe proof of Theorem 2 is completed.
Note 1. Lemma 4 and Theorem 2 show that the bounded absorbing sets are compact bounded absorbing sets in . Therefore, using condition 3 in Lemma 2, the operator semigroup only needs to be a continuous operator; from Theorem 1, it can be seen that the semigroup is already a continuous semigroup; from this angle, the family of global attractors of problems (1)–(3) in can also be obtained.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
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).