June  2011, 31(2): 445-467. doi: 10.3934/dcds.2011.31.445

Exponential attractors for lattice dynamical systems in weighted spaces

1. 

221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849

Received  June 2010 Revised  February 2011 Published  June 2011

We first present some sufficient conditions for the existence of exponential attractors for locally coupled lattice dynamical systems in weighted spaces of infinite sequences. Then we apply this result to discuss the existence of exponential attractors for first order lattice systems, partly dissipative lattice systems, and second order lattice systems in weighted spaces of infinite sequences.
Citation: Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445
References:
[1]

A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems,, J. Math. Anal. Appl., 339 (2008), 217.  doi: 10.1016/j.jmaa.2007.06.054.  Google Scholar

[2]

A. Y. Abdallah, Exponential attractors for second order lattice dynamical systems,, Commun. Pure Appl. Anal., 8 (2009), 803.  doi: 10.3934/cpaa.2009.8.803.  Google Scholar

[3]

A. V. Babin and B. Nicolaenko, Exponential attractors for reaction diffusion equations in unbounded domains,, J. Dyna. Diff. Eqs., 7 (1995), 567.  doi: 10.1007/BF02218725.  Google Scholar

[4]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, Internat. J. Bifurcation and Choas, 11 (2001), 143.  doi: 10.1142/S0218127401002031.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).   Google Scholar

[6]

S. N. Chow, Lattice dynamical systems,, in, (2003), 1.   Google Scholar

[7]

D. N. Cheban and D. S. Fakeeh, Global attractors of infinite-dimensional dynamical systems, III,, Bull. Acad. Sci. Rep. Moldova Mat., 18-19 (1995), 18.   Google Scholar

[8]

Z. Dai and D. Ma, Exponential attractors of the nonlinear wave equations,, Chinese Science Bulletin, 43 (1998), 1331.  doi: 10.1007/BF02883676.  Google Scholar

[9]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces,, J. Dyn. Diff. Eqs., 13 (2001), 791.  doi: 10.1023/A:1016676027666.  Google Scholar

[10]

A. Eden, C. Foias and V. Kalantarov, A remark on two constructions of exponential attractors for $\alpha $-contractions,, J. Dyn. Diff. Eqs., 10 (1998), 37.  doi: 10.1023/A:1022636328133.  Google Scholar

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equation,", Research in Applied Mathematics 37, (1994).   Google Scholar

[12]

X. Fan, Exponential attractors for a first-order dissipative lattice dynamical systems,, J. Appl. Math., 2008 (2008), 1.  doi: 10.1155/2008/354652.  Google Scholar

[13]

X. Fan and H. Yang, Exponential attractor and its fractal dimension for a second order lattice dynamical system,, J. Math. Anal. Appl., 367 (2010), 350.  doi: 10.1016/j.jmaa.2009.11.003.  Google Scholar

[14]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).   Google Scholar

[15]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation,, J. Differential Equations, 217 (2005), 88.  doi: 10.1016/j.jde.2005.06.002.  Google Scholar

[16]

X. Li and D. Wang, Attractors for partly dissipative lattice dynamic systems in weighted spaces,, J. Math. Anal. Appl., 325 (2007), 141.  doi: 10.1016/j.jmaa.2006.01.054.  Google Scholar

[17]

X. Li and C. Zhong, Attractors for partly dissipative lattice dynamic systems in $l^2\times l^2$,, J. Computational and Applied Mathematics, 177 (2005), 159.  doi: 10.1016/j.cam.2004.09.014.  Google Scholar

[18]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1997).   Google Scholar

[19]

B. Wang, Dynamics of systems on infinite lattices,, J. Differential Equations, 221 (2006), 224.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[20]

C. Zhao and S. Zhou, Sufficient conditions for the existence of exponential attractors for lattice systems and applications,, Acta Math. Sinica, 53 (2010), 233.   Google Scholar

[21]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems,, Disc. Cont. Dyn. Syst., 9 (2008), 763.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[22]

S. Zhou, Attractors for second order lattice dynamical systems,, J. Differential Equations, 179 (2002), 605.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[23]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems,, J. Differential Equations, 224 (2006), 172.  doi: 10.1016/j.jde.2005.06.024.  Google Scholar

show all references

References:
[1]

A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems,, J. Math. Anal. Appl., 339 (2008), 217.  doi: 10.1016/j.jmaa.2007.06.054.  Google Scholar

[2]

A. Y. Abdallah, Exponential attractors for second order lattice dynamical systems,, Commun. Pure Appl. Anal., 8 (2009), 803.  doi: 10.3934/cpaa.2009.8.803.  Google Scholar

[3]

A. V. Babin and B. Nicolaenko, Exponential attractors for reaction diffusion equations in unbounded domains,, J. Dyna. Diff. Eqs., 7 (1995), 567.  doi: 10.1007/BF02218725.  Google Scholar

[4]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, Internat. J. Bifurcation and Choas, 11 (2001), 143.  doi: 10.1142/S0218127401002031.  Google Scholar

[5]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).   Google Scholar

[6]

S. N. Chow, Lattice dynamical systems,, in, (2003), 1.   Google Scholar

[7]

D. N. Cheban and D. S. Fakeeh, Global attractors of infinite-dimensional dynamical systems, III,, Bull. Acad. Sci. Rep. Moldova Mat., 18-19 (1995), 18.   Google Scholar

[8]

Z. Dai and D. Ma, Exponential attractors of the nonlinear wave equations,, Chinese Science Bulletin, 43 (1998), 1331.  doi: 10.1007/BF02883676.  Google Scholar

[9]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces,, J. Dyn. Diff. Eqs., 13 (2001), 791.  doi: 10.1023/A:1016676027666.  Google Scholar

[10]

A. Eden, C. Foias and V. Kalantarov, A remark on two constructions of exponential attractors for $\alpha $-contractions,, J. Dyn. Diff. Eqs., 10 (1998), 37.  doi: 10.1023/A:1022636328133.  Google Scholar

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equation,", Research in Applied Mathematics 37, (1994).   Google Scholar

[12]

X. Fan, Exponential attractors for a first-order dissipative lattice dynamical systems,, J. Appl. Math., 2008 (2008), 1.  doi: 10.1155/2008/354652.  Google Scholar

[13]

X. Fan and H. Yang, Exponential attractor and its fractal dimension for a second order lattice dynamical system,, J. Math. Anal. Appl., 367 (2010), 350.  doi: 10.1016/j.jmaa.2009.11.003.  Google Scholar

[14]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).   Google Scholar

[15]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation,, J. Differential Equations, 217 (2005), 88.  doi: 10.1016/j.jde.2005.06.002.  Google Scholar

[16]

X. Li and D. Wang, Attractors for partly dissipative lattice dynamic systems in weighted spaces,, J. Math. Anal. Appl., 325 (2007), 141.  doi: 10.1016/j.jmaa.2006.01.054.  Google Scholar

[17]

X. Li and C. Zhong, Attractors for partly dissipative lattice dynamic systems in $l^2\times l^2$,, J. Computational and Applied Mathematics, 177 (2005), 159.  doi: 10.1016/j.cam.2004.09.014.  Google Scholar

[18]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1997).   Google Scholar

[19]

B. Wang, Dynamics of systems on infinite lattices,, J. Differential Equations, 221 (2006), 224.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[20]

C. Zhao and S. Zhou, Sufficient conditions for the existence of exponential attractors for lattice systems and applications,, Acta Math. Sinica, 53 (2010), 233.   Google Scholar

[21]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems,, Disc. Cont. Dyn. Syst., 9 (2008), 763.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[22]

S. Zhou, Attractors for second order lattice dynamical systems,, J. Differential Equations, 179 (2002), 605.  doi: 10.1006/jdeq.2001.4032.  Google Scholar

[23]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems,, J. Differential Equations, 224 (2006), 172.  doi: 10.1016/j.jde.2005.06.024.  Google Scholar

[1]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[2]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803

[3]

Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019

[4]

Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019218

[5]

Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085

[6]

Fang-Di Dong, Wan-Tong Li, Li Zhang. Entire solutions in a two-dimensional nonlocal lattice dynamical system. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2517-2545. doi: 10.3934/cpaa.2018120

[7]

Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329

[8]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

[9]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197

[10]

Chin-Chin Wu. Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2813-2827. doi: 10.3934/dcds.2017121

[11]

Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure & Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038

[12]

Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias, Mai Zurwatul Ahlam Mohd Jaffar. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 89-94. doi: 10.3934/naco.2017005

[13]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[14]

Rogério Martins. One-dimensional attractor for a dissipative system with a cylindrical phase space. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 533-547. doi: 10.3934/dcds.2006.14.533

[15]

Chengjian Zhang, Lu Zhao. The attractors for 2nd-order stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 575-590. doi: 10.3934/dcds.2017023

[16]

Daniele Mundici. The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 537-549. doi: 10.3934/dcds.2008.21.537

[17]

Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339

[18]

Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107

[19]

Chiun-Chuan Chen, Ting-Yang Hsiao, Li-Chang Hung. Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 153-187. doi: 10.3934/dcds.2020007

[20]

George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]