American Institute of Mathematical Sciences

Advanced
January  2005, 12(1): 27-38. doi: 10.3934/dcds.2005.12.27

On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging

V. V. Chepyzhov 1, , M. I. Vishik 2, and  W. L. Wendland 3,

1. 

Institute for Problems of Information Transmission, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 101447, GSP-4, Russian Federation
2. 

Institute for Problems of Information Transmission, Russian Academy of Sciences, 101 447 Moscow GSP-4, Russian Federation
3. 

Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, 70550 Stuttgart, Germany

Received  August 2003 Revised  December 2003 Published  December 2004

We study the global attractors for the dissipative sine--Gordon type wave equation with time dependent external force $g(x,t)$. We assume that the function $g(x,t)$ is translationary compact in $L^{l o c}_2(\mathbb R,L_2 (\Omega))$ and the nonlinear function $f(u)$ is bounded and satisfies a global Lipschitz condition. If the Lipschitz constant $K$ is smaller than the first eigenvalue of the Laplacian with homogeneous Dirichlet conditions and the dissipation coefficient is large, then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the wave equation. Moreover, the attractor attracts all the solutions of the equation with exponential rate.
We also consider the wave equation with rapidly oscillating external force $g^\varepsilon(x,t)=g(x,t,t/\varepsilon)$ having the average $g^0(x,t)$ as $\varepsilon\to 0+$. We assume that the function $g(x,t,\zeta)-g^0(x,t)$ has a bounded primitive with respect to $\zeta$. Then we prove that the Hausdorff distance between the global attractor $\mathcal A_\varepsilon$ of the original equation and the global attractor $\mathcal A_0$ of the averaged equation is less than $O(\varepsilon^{1/2})$.
Keywords: time averaging., dissipative wave equation, Sine-Gordon equation, global attractor.
Mathematics Subject Classification: 35B40, 35L70, 34D45, 34C2.
Citation: V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27
[1]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925
[2]

Goong Chen, Zhonghai Ding, Shujie Li. On positive solutions of the elliptic sine-Gordon equation. Communications on Pure & Applied Analysis, 2005, 4 (2) : 283-294. doi: 10.3934/cpaa.2005.4.283
[3]

Qin Sheng, David A. Voss, Q. M. Khaliq. An adaptive splitting algorithm for the sine-Gordon equation. Conference Publications, 2005, 2005 (Special) : 792-797. doi: 10.3934/proc.2005.2005.792
[4]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025
[5]

Cornelia Schiebold. Noncommutative AKNS systems and multisoliton solutions to the matrix sine-gordon equation. Conference Publications, 2009, 2009 (Special) : 678-690. doi: 10.3934/proc.2009.2009.678
[6]

Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915
[7]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082
[8]

Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939
[9]

Boling Guo, Zhengde Dai. Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 783-793. doi: 10.3934/dcds.1998.4.783
[10]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[11]

Sara Cuenda, Niurka R. Quintero, Angel Sánchez. Sine-Gordon wobbles through Bäcklund transformations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1047-1056. doi: 10.3934/dcdss.2011.4.1047
[12]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094
[13]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410
[14]

Ivan Christov, C. I. Christov. The coarse-grain description of interacting sine-Gordon solitons with varying widths. Conference Publications, 2009, 2009 (Special) : 171-180. doi: 10.3934/proc.2009.2009.171
[15]

Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056
[16]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407
[17]

Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems & Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034
[18]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179
[19]

Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217
[20]

Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (34)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences