波方程与欧拉伯努利板方程耦合系统的全局吸引子

Abstract

Abstract:

In this paper, we consider the longtime behavior for a coupled system consisting of the semi-linear wave equation with nonlinear boundary dissipation and the Euler-Bernoulli plate equation on a Riemannian manifold. It is shown that the existence of global and compact attractors depends on the curvature properties of the metric on the manifold by using the multiplier method and the hypothesis of escape vector field.

Key words: Global attractor, Coupled wave/plate equation, Geometric multiplier method, Nonlinear boundary dissipation

CLC Number:

O231.4

Peng Qingqing,Zhang Zhifei. Global Attractor for a Coupled System of Wave and Euler-Bernoulli Plate Equation with Boundary Weak Damping[J].Acta mathematica scientia,Series A, 2023, 43(4): 1179-1196.

Trendmd

References 27

[1]	Aliev A B, Farhadova Y M. Existence of global attractors for the coupled system of suspension bridge equations. Azerbaijan Journal of Mathematics, 2021, 11(2): 105-124
[2]	Barbu V. Nonlinear Semigroups and Differential Equations in Banach Spaces. Bucuresti: Editura Academiei, 1976
[3]	Bucci F, Chueshov I, Lasiecka I. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure and Applied Analysis, 2007, 6: 113-140 doi: 10.3934/cpaa.2007.6.113
[4]	Bucci F, Chueshov I. Long time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Arxiv preprint arxiv: 0806.4500, 2008
[5]	Chen B Y, Zhao C X, Zhong C K. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems-B, 2021, 26(12): 6207-6228 doi: 10.3934/dcdsb.2021015
[6]	Chueshov I, Eller M, Lasiecka I. On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun PDE, 2002, 27(9/10): 1901-1951 doi: 10.1081/PDE-120016132
[7]	Chueshov I, Eller M, Lasiecka I. Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation. Communications in Partial Differential Equations, 2005, 29(11/12): 1847-1876 doi: 10.1081/PDE-200040203
[8]	Chueshov I, Lasiecka I. Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping. Providence: Memoirs of the American Mathematical Society, 2008
[9]	Feireisl E. Global attractors for semilinear damped wave equations with supercritical exponent. Journal of Differential Equations, 1995, 116(2): 431-447 doi: 10.1006/jdeq.1995.1042
[10]	Ghidaglia J M, Temam R. Regularity of the solutions of second order evolution equations and their attractors. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1987, 14(3): 485-511
[11]	Grisvard P. Caractérisation de quelques espaces interpolation. Archive for Rational Mechanics 1967, 25(1): 40-63
[12]	Guo Y X, Yao P F. Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback. Journal of Mathematical Analysis and Applications, 2006, 317(1): 50-70 doi: 10.1016/j.jmaa.2005.12.006
[13]	Hale J K. Asymptotic Behavior of Dissipative Systems. Providence: American Mathematical Soc, 2010
[14]	Haraux A. Two remarks on dissipative hyperbolic problems. Research Notes in Mathematics, 1985, 122: 161-179
[15]	Lasiecka I, Triggiani R. Uniform stabilization of the wave equation with dirichlet or neumann feedback control without geometrical conditions. Applied Mathematics and Optimization, 1992, 25(2): 189-224 doi: 10.1007/BF01182480
[16]	Lasiecka I, Triggiani R, Yao P F. Carleman estimates for a plate equation on a riemann manifold with energy level terms. Analysis and Applications-ISAAC, 2001, 2003: 199-236
[17]	Lions J, Magenes E. Non-Homogeneous Boundary Value Problems and Applications. Berlin: Springer Science and Business Media, 2012
[18]	Qin Y M. Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors. Berlin: Springer Science and Business Media, 2008
[19]	Simon J. Compact sets in the space $L^p([0,T];B)$. Annali di Matematica pura ed applicata, 1986, 146(1): 65-96 doi: 10.1007/BF01762360
[20]	Song W J, Yang G S. Existence of the global attractor to fractional order generalized coupled nonlinear schrodinger equations with derivative. Boundary Value Problems, 2018, 109(2018): 1-27
[21]	Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Berlin: Springer Science and Business Media, 2012
[22]	Triggiani R, Yao P F. Carleman estimates with no lower-order terms for general riemann wave equations. global uniqueness and observability in one shot. Applied Mathematics and Optimization, 2002, 46(2): 331-375 doi: 10.1007/s00245-002-0751-5
[23]	Wang J C, Yao P F. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure and Applied Analysis, 2021, 21(6): 1-15
[24]	Wu H, Shen C L, Yu Y L. An Introduction to Riemannian Geometry. Beijing: Univ of Beijing, 1989
[25]	Yao P F. Observability inequalities for shallow shells. SIAM Journal on Control and Optimization, 2000, 38(6): 1729-1756 doi: 10.1137/S0363012999338692
[26]	Yao P F. Modeling and Control in Vibrational and Structural Dynamics:a Differential Geometric Approach. Boca Raton: CRC Press, 2011
[27]	Zhao C X, Zhao C Y, Zhong C K. The global attractor for a class of extensible beams with nonlocal weak damping. Discrete and Continuous Dynamical Systems-B, 2020, 25(3): 935-955 doi: 10.3934/dcdsb.2019197

Global Attractor for a Coupled System of Wave and Euler-Bernoulli Plate Equation with Boundary Weak Damping

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 27

Related Articles 14

Metrics

Comments

Recommended 10

[1]	Zhu Kaixuan, Sun Tao, Xie Yongqin. Global Attractors for a Class of Reaction-Diffusion Equations with Distribution Derivatives in $\Bbb R ^{n}$ [J]. Acta mathematica scientia,Series A, 2023, 43(1): 82-92.
[2]	Xiangyu Xiao,Zhilin Pu. The Global Attractors of Cahn-Hilliard-Brinkman System [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1027-1040.
[3]	Didi Hu,Xuan Wang. The Strong Time-Dependent Global Attractors for the Non-Damping Abstract Evolution Equations with Fading Memory [J]. Acta mathematica scientia,Series A, 2019, 39(1): 81-94.
[4]	Xiaoming Peng,Xiaoxiao Zheng,Yadong Shang. Global Attractors for a Fourth Order Pseudo-Parabolic Equation with Fading Memory [J]. Acta mathematica scientia,Series A, 2019, 39(1): 114-124.
[5]	Xuan Wang, Ying Han, Chenghua Gao. Global Attractor in H¹（Rⁿ) x L_u²(R⁺;H¹(Rⁿ)) for the Nonclassical Diffusion Equations with Fading Memory [J]. Acta mathematica scientia,Series A, 2018, 38(6): 1205-1223.
[6]	Liu Shifang, Ma Qiaozhen. Existence of Strong Global Attractors for Damped Suspension Bridge Equations with History Memory [J]. Acta mathematica scientia,Series A, 2017, 37(4): 684-697.
[7]	Geng Fan, Li Ruizhai, Ge Xiangyu. Long Time Behavior of Boussinesq Type Equation with Damping Term [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1196-1210.
[8]	Ma Qiaozhen, Xu Ling, Zhang Yanjun. Asymptotic Behavior of the Solution for the Nonclassical Diffusion Equations with Fading Memory on the Whole Space R³ [J]. Acta mathematica scientia,Series A, 2016, 36(1): 36-48.
[9]	Zhang Yanjun, Ma Qiaozhen. Attractors for the Nonclassical Diffusion Equations with Critical Nonlinearity on the Whole Space R³ [J]. Acta mathematica scientia,Series A, 2015, 35(2): 294-305.
[10]	MA Wen-Jun, MA Qiao-Zhen, GAO Pei-Ming. Regularity and Global Attractor of Nonlinear Elastic Rod Oscillation Equation [J]. Acta mathematica scientia,Series A, 2014, 34(2): 445-453.
[11]	YANG Zhi-Jian, CHENG Jian-Ling. Asymptotic Behavior of Solutions to the Kirchhoff Type Equation [J]. Acta mathematica scientia,Series A, 2011, 31(4): 1008-1021.
[12]	Ma Qiaozhen; Sun Chunyou; Zhong Chengkui. Existence of Strong Global Attractors for the Nonlinear Beam Equations [J]. Acta mathematica scientia,Series A, 2007, 27(5): 941-948.
[13]	LI Dong-Long, GUO Bai-Ling. Global Attractor for Complex Ginzburg Landau\=Equation in Whole R\+3 [J]. Acta mathematica scientia,Series A, 2004, 4(5): 607-617.
[14]	SHANG Ya-Dong, GUO Bo-Ling. The Global Attractors for the Periodic Initial Value Problem for Dissipative Generalized Symmetric Regularized Long Wave Equations [J]. Acta mathematica scientia,Series A, 2003, 23(6): 745-757.