# American Institute of Mathematical Sciences

October  2015, 20(8): 2715-2732. doi: 10.3934/dcdsb.2015.20.2715

## Long time dynamics of a multidimensional nonlinear lattice with memory

 1 Department of Mathematics, Federal University of Santa Catarina, Florianópolis, S.C. 88040-900, Brazil, Brazil 2 National Laboratory of Scientific Computation, CEP 25651-070, Petrópolis, RJ, Brazil

Received  November 2014 Revised  May 2015 Published  August 2015

This work is devoted to study the nature of vibrations arising in a multidimensional nonlinear periodic lattice structure with memory. We prove the existence of a global attractor. In the homogeneous case under a restriction on the nonlinear term we obtain decay rates of the total energy. These rates could be exponential, polynomial or several other intermediate types.
Citation: Jáuber Cavalcante Oliveira, Jardel Morais Pereira, Gustavo Perla Menzala. Long time dynamics of a multidimensional nonlinear lattice with memory. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2715-2732. doi: 10.3934/dcdsb.2015.20.2715
##### References:
 [1] A. Y. Abdallah, Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems,, Comm. on Pure and Applied Analysis, 5 (2006), 55. doi: 10.3934/cpaa.2006.5.55. Google Scholar [2] W. L. Briggs and V. E. Henson, The DFT, an Owner's Manual for the Discrete Fourier Transform,, SIAM, (1995). Google Scholar [3] T. Chen, S. Zhou and C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay,, Acta Mathematicae Appl. Sinica, 26 (2010), 633. doi: 10.1007/s10255-007-7101-y. Google Scholar [4] L. O. Chua and T. Roska, The CNN paradigma,, IEEE Trans. Circuits Systems, 40 (1993), 147. Google Scholar [5] J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford University Press, (1985). Google Scholar [6] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces,, Discrete and Continuous Dynamical Systems, 31 (2011), 445. doi: 10.3934/dcds.2011.31.445. Google Scholar [7] R. Hirota and J. Satsuma, N-solution of nonlinear network equations describing a Volterra system,, J. Phys. Soc. Japan., 40 (1976), 891. doi: 10.1143/JPSJ.40.891. Google Scholar [8] N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation,, Journal of Differential Equations, 217 (2005), 88. doi: 10.1016/j.jde.2005.06.002. Google Scholar [9] R. Kapral, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113. doi: 10.1007/BF01192578. Google Scholar [10] N. I. Karachalios and A. N. Yannacopoulos, The existence of a global attractor for the discrete nonlinear Schrödinger equation. II. Compacteness without tail estimates in $\mathbbZ^N, N\geq 1$, lattices,, Proc. Royal Soc. of Edinburgh, 137 (2007), 63. doi: 10.1017/S0308210505000831. Google Scholar [11] V. V. Konotop and G. Perla Menzala, Localized solutions of a nonlinear diatomic lattice,, Quarterly of Applied Mathematics, 63 (2005), 201. doi: 10.1090/S0033-569X-05-00952-6. Google Scholar [12] V. V. Konotop, J. M. Rivera and G. Perla Menzala, Uniform rates of decay of solutions for a nonlinear lattice with memory,, Asymptotic Analysis, 38 (2004), 167. Google Scholar [13] S. A. Messaoudi, General decay of solutions of a viscoelastic equation,, J. Math. Anal., 341 (2008), 1457. doi: 10.1016/j.jmaa.2007.11.048. Google Scholar [14] S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source,, Nonlinear Analysis, 69 (2008), 2589. doi: 10.1016/j.na.2007.08.035. Google Scholar [15] J. C. Oliveira, J. M. Pereira and G. Perla Menzala, Attractors for second order periodic lattices with nonlinear damping,, Journal of Difference Equations and Applications, 14 (2008), 899. doi: 10.1080/10236190701859211. Google Scholar [16] J. C. Oliveira, J. M. Pereira and G. Perla Menzala, Large time behavior of multidimensional nonlinear lattices with nonlinear damping,, Communications in Applied Analysis, 14 (2010), 155. Google Scholar [17] A. Perez-Muñuzuri, V. Perez-Mañuzuri, V. Perez-Villar and L. O. Chua, Spiral waves on a 2-D array of nonlinear circuits,, IEEE Trans. Circuits Systems, 40 (1993), 872. doi: 10.1109/81.251828. Google Scholar [18] R. Racke and C. Shang, Global attractors for nonlinear beam equations,, Proceedings of the Royal Society of Edinburgh, 142 (2012), 1087. doi: 10.1017/S030821051000168X. Google Scholar [19] M. A. J. Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory,, Journal of Mathematical Physics, 54 (2013). doi: 10.1063/1.4792606. Google Scholar [20] R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [21] J. von Neumann, The general and logical theory of automata,, in Cerebral Mechanisms in Behavior (ed. L. A. Jeffress), (1951), 9. Google Scholar [22] B. Wang, Dynamics of systems on infinite lattices,, Journal of Differential Equations, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. Google Scholar [23] Y. Yan, Attractors and dimensions for discretization of a weakly damped Schrodinger equation and a Sine-Gordon equation,, Nonlinear Analysis TMA, 20 (1993), 1417. doi: 10.1016/0362-546X(93)90168-R. Google Scholar [24] V. E. Zakharov, S. L. Musher and A. M. Rubenchik, Nonlinear stage of parametric wave excitation in a plasma,, Sov. Phys. JETP, 19 (1974), 151. Google Scholar [25] S. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. Google Scholar [26] 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, Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems,, Comm. on Pure and Applied Analysis, 5 (2006), 55. doi: 10.3934/cpaa.2006.5.55. Google Scholar [2] W. L. Briggs and V. E. Henson, The DFT, an Owner's Manual for the Discrete Fourier Transform,, SIAM, (1995). Google Scholar [3] T. Chen, S. Zhou and C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay,, Acta Mathematicae Appl. Sinica, 26 (2010), 633. doi: 10.1007/s10255-007-7101-y. Google Scholar [4] L. O. Chua and T. Roska, The CNN paradigma,, IEEE Trans. Circuits Systems, 40 (1993), 147. Google Scholar [5] J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford University Press, (1985). Google Scholar [6] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces,, Discrete and Continuous Dynamical Systems, 31 (2011), 445. doi: 10.3934/dcds.2011.31.445. Google Scholar [7] R. Hirota and J. Satsuma, N-solution of nonlinear network equations describing a Volterra system,, J. Phys. Soc. Japan., 40 (1976), 891. doi: 10.1143/JPSJ.40.891. Google Scholar [8] N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation,, Journal of Differential Equations, 217 (2005), 88. doi: 10.1016/j.jde.2005.06.002. Google Scholar [9] R. Kapral, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113. doi: 10.1007/BF01192578. Google Scholar [10] N. I. Karachalios and A. N. Yannacopoulos, The existence of a global attractor for the discrete nonlinear Schrödinger equation. II. Compacteness without tail estimates in $\mathbbZ^N, N\geq 1$, lattices,, Proc. Royal Soc. of Edinburgh, 137 (2007), 63. doi: 10.1017/S0308210505000831. Google Scholar [11] V. V. Konotop and G. Perla Menzala, Localized solutions of a nonlinear diatomic lattice,, Quarterly of Applied Mathematics, 63 (2005), 201. doi: 10.1090/S0033-569X-05-00952-6. Google Scholar [12] V. V. Konotop, J. M. Rivera and G. Perla Menzala, Uniform rates of decay of solutions for a nonlinear lattice with memory,, Asymptotic Analysis, 38 (2004), 167. Google Scholar [13] S. A. Messaoudi, General decay of solutions of a viscoelastic equation,, J. Math. Anal., 341 (2008), 1457. doi: 10.1016/j.jmaa.2007.11.048. Google Scholar [14] S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source,, Nonlinear Analysis, 69 (2008), 2589. doi: 10.1016/j.na.2007.08.035. Google Scholar [15] J. C. Oliveira, J. M. Pereira and G. Perla Menzala, Attractors for second order periodic lattices with nonlinear damping,, Journal of Difference Equations and Applications, 14 (2008), 899. doi: 10.1080/10236190701859211. Google Scholar [16] J. C. Oliveira, J. M. Pereira and G. Perla Menzala, Large time behavior of multidimensional nonlinear lattices with nonlinear damping,, Communications in Applied Analysis, 14 (2010), 155. Google Scholar [17] A. Perez-Muñuzuri, V. Perez-Mañuzuri, V. Perez-Villar and L. O. Chua, Spiral waves on a 2-D array of nonlinear circuits,, IEEE Trans. Circuits Systems, 40 (1993), 872. doi: 10.1109/81.251828. Google Scholar [18] R. Racke and C. Shang, Global attractors for nonlinear beam equations,, Proceedings of the Royal Society of Edinburgh, 142 (2012), 1087. doi: 10.1017/S030821051000168X. Google Scholar [19] M. A. J. Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory,, Journal of Mathematical Physics, 54 (2013). doi: 10.1063/1.4792606. Google Scholar [20] R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [21] J. von Neumann, The general and logical theory of automata,, in Cerebral Mechanisms in Behavior (ed. L. A. Jeffress), (1951), 9. Google Scholar [22] B. Wang, Dynamics of systems on infinite lattices,, Journal of Differential Equations, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. Google Scholar [23] Y. Yan, Attractors and dimensions for discretization of a weakly damped Schrodinger equation and a Sine-Gordon equation,, Nonlinear Analysis TMA, 20 (1993), 1417. doi: 10.1016/0362-546X(93)90168-R. Google Scholar [24] V. E. Zakharov, S. L. Musher and A. M. Rubenchik, Nonlinear stage of parametric wave excitation in a plasma,, Sov. Phys. JETP, 19 (1974), 151. Google Scholar [25] S. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. Google Scholar [26] 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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