\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Long time dynamics of a multidimensional nonlinear lattice with memory

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 34D45, 34D05; Secondary: 39A12.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(86) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return