# American Institute of Mathematical Sciences

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

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##### 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
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