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 and Continuous Dynamical Systems, 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), 217-224. doi: 10.1016/j.jmaa.2007.06.054.

[2]

A. Y. Abdallah, Exponential attractors for second order lattice dynamical systems, Commun. Pure Appl. Anal., 8 (2009), 803-813. doi: 10.3934/cpaa.2009.8.803.

[3]

A. V. Babin and B. Nicolaenko, Exponential attractors for reaction diffusion equations in unbounded domains, J. Dyna. Diff. Eqs., 7 (1995), 567-590. doi: 10.1007/BF02218725.

[4]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifurcation and Choas, 11 (2001), 143-153. doi: 10.1142/S0218127401002031.

[5]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[6]

S. N. Chow, Lattice dynamical systems, in "Dynamical Systems," Lecture Notes in Math., 1822, Springer, Berlin, (2003), 1-102.

[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), 3-13.

[8]

Z. Dai and D. Ma, Exponential attractors of the nonlinear wave equations, Chinese Science Bulletin, 43 (1998), 1331-1335. doi: 10.1007/BF02883676.

[9]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dyn. Diff. Eqs., 13 (2001) 791-806. doi: 10.1023/A:1016676027666.

[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), 37-45. doi: 10.1023/A:1022636328133.

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equation," Research in Applied Mathematics 37, Masson/John Wiley co-publication, Paris, 1994.

[12]

X. Fan, Exponential attractors for a first-order dissipative lattice dynamical systems, J. Appl. Math., 2008 (2008), 1-8. doi: 10.1155/2008/354652.

[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), 350-359. doi: 10.1016/j.jmaa.2009.11.003.

[14]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI. 1988,

[15]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123. doi: 10.1016/j.jde.2005.06.002.

[16]

X. Li and D. Wang, Attractors for partly dissipative lattice dynamic systems in weighted spaces, J. Math. Anal. Appl., 325 (2007), 141-156. doi: 10.1016/j.jmaa.2006.01.054.

[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), 159-174. doi: 10.1016/j.cam.2004.09.014.

[18]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 2nd ed., 1997.

[19]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.

[20]

C. Zhao and S. Zhou, Sufficient conditions for the existence of exponential attractors for lattice systems and applications, Acta Math. Sinica, 53 (2010), 233-242.

[21]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Disc. Cont. Dyn. Syst., 9 (2008), 763-785. doi: 10.3934/dcdsb.2008.9.763.

[22]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.

[23]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.

show all references

References:
[1]

A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224. doi: 10.1016/j.jmaa.2007.06.054.

[2]

A. Y. Abdallah, Exponential attractors for second order lattice dynamical systems, Commun. Pure Appl. Anal., 8 (2009), 803-813. doi: 10.3934/cpaa.2009.8.803.

[3]

A. V. Babin and B. Nicolaenko, Exponential attractors for reaction diffusion equations in unbounded domains, J. Dyna. Diff. Eqs., 7 (1995), 567-590. doi: 10.1007/BF02218725.

[4]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifurcation and Choas, 11 (2001), 143-153. doi: 10.1142/S0218127401002031.

[5]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[6]

S. N. Chow, Lattice dynamical systems, in "Dynamical Systems," Lecture Notes in Math., 1822, Springer, Berlin, (2003), 1-102.

[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), 3-13.

[8]

Z. Dai and D. Ma, Exponential attractors of the nonlinear wave equations, Chinese Science Bulletin, 43 (1998), 1331-1335. doi: 10.1007/BF02883676.

[9]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, J. Dyn. Diff. Eqs., 13 (2001) 791-806. doi: 10.1023/A:1016676027666.

[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), 37-45. doi: 10.1023/A:1022636328133.

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equation," Research in Applied Mathematics 37, Masson/John Wiley co-publication, Paris, 1994.

[12]

X. Fan, Exponential attractors for a first-order dissipative lattice dynamical systems, J. Appl. Math., 2008 (2008), 1-8. doi: 10.1155/2008/354652.

[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), 350-359. doi: 10.1016/j.jmaa.2009.11.003.

[14]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, RI. 1988,

[15]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123. doi: 10.1016/j.jde.2005.06.002.

[16]

X. Li and D. Wang, Attractors for partly dissipative lattice dynamic systems in weighted spaces, J. Math. Anal. Appl., 325 (2007), 141-156. doi: 10.1016/j.jmaa.2006.01.054.

[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), 159-174. doi: 10.1016/j.cam.2004.09.014.

[18]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 2nd ed., 1997.

[19]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.

[20]

C. Zhao and S. Zhou, Sufficient conditions for the existence of exponential attractors for lattice systems and applications, Acta Math. Sinica, 53 (2010), 233-242.

[21]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Disc. Cont. Dyn. Syst., 9 (2008), 763-785. doi: 10.3934/dcdsb.2008.9.763.

[22]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.

[23]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.

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