Long time dynamics of a multidimensional nonlinear lattice with memory

Jáuber Cavalcante  Oliveira; Jardel Morais  Pereira; Gustavo Perla  Menzala

doi:10.3934/dcdsb.2015.20.2715

Article Contents

2015, Volume 20, Issue 8: 2715-2732. Doi: 10.3934/dcdsb.2015.20.2715

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Long time dynamics of a multidimensional nonlinear lattice with memory

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

Received: November 2014

Revised: May 2015

Published: October 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 34D45, 34D05; Secondary: 39A12.

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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), 55-69.doi: 10.3934/cpaa.2006.5.55.
[2]	W. L. Briggs and V. E. Henson, The DFT, an Owner's Manual for the Discrete Fourier Transform, SIAM, Philadelphia, 1995.
[3]	T. Chen, S. Zhou and C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay, Acta Mathematicae Appl. Sinica, English Series, 26 (2010), 633-642.doi: 10.1007/s10255-007-7101-y.
[4]	L. O. Chua and T. Roska, The CNN paradigma, IEEE Trans. Circuits Systems, 40 (1993), 147-156. Available from: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=222795.
[5]	J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, NY, 1985.
[6]	X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete and Continuous Dynamical Systems, 31 (2011), 445-467.doi: 10.3934/dcds.2011.31.445.
[7]	R. Hirota and J. Satsuma, N-solution of nonlinear network equations describing a Volterra system, J. Phys. Soc. Japan., 40 (1976), 891-900.doi: 10.1143/JPSJ.40.891.
[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), 88-123.doi: 10.1016/j.jde.2005.06.002.
[9]	R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.doi: 10.1007/BF01192578.
[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), 63-76.doi: 10.1017/S0308210505000831.
[11]	V. V. Konotop and G. Perla Menzala, Localized solutions of a nonlinear diatomic lattice, Quarterly of Applied Mathematics, 63 (2005), 201-223.doi: 10.1090/S0033-569X-05-00952-6.
[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), 167-185.
[13]	S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal., 341 (2008), 1457-1467.doi: 10.1016/j.jmaa.2007.11.048.
[14]	S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis, 69 (2008), 2589-2598.doi: 10.1016/j.na.2007.08.035.
[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), 899-921.doi: 10.1080/10236190701859211.
[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), 155-176.
[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), 872-877.doi: 10.1109/81.251828.
[18]	R. Racke and C. Shang, Global attractors for nonlinear beam equations, Proceedings of the Royal Society of Edinburgh, 142 (2012), 1087-1107.doi: 10.1017/S030821051000168X.
[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), 021505, 15pp.doi: 10.1063/1.4792606.
[20]	R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988.doi: 10.1007/978-1-4684-0313-8.
[21]	J. von Neumann, The general and logical theory of automata, in Cerebral Mechanisms in Behavior (ed. L. A. Jeffress), Wiley, New York, 1951, 9-31.
[22]	B. Wang, Dynamics of systems on infinite lattices, Journal of Differential Equations, 221 (2006), 224-245.doi: 10.1016/j.jde.2005.01.003.
[23]	Y. Yan, Attractors and dimensions for discretization of a weakly damped Schrodinger equation and a Sine-Gordon equation, Nonlinear Analysis TMA, 20 (1993), 1417-1452.doi: 10.1016/0362-546X(93)90168-R.
[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), 151-152. Available from: http://jetpletters.ac.ru/ps/1774/article_26980.shtml.
[25]	S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.doi: 10.1016/j.jde.2004.02.005.
[26]	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.