Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation

Pages: 711 - 736, Volume 19, Issue 4, December 2007      doi:10.3934/dcds.2007.19.711

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N. I. Karachalios - Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece (email)
Hector E. Nistazakis - Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi GR 83200, Samos, Greece (email)
Athanasios N. Yannacopoulos - Department of Statistics, Athens University of Economics and Business, Patission 76 GR 10434, Athens, Greece (email)

Abstract: We study the asymptotic behavior of complex discrete evolution equations of Ginzburg- Landau type. Depending on the nonlinearity and the data of the problem, we find different dynamical behavior ranging from global existence of solutions and global attractors to blow-up in finite time. We provide estimates for the blow-up time, depending not only on the initial data but also on the size of the lattice. Some of the theoretical results are tested by numerical simulations.

Keywords:  Discrete Ginzburg-Landau equation, lattice differential equations, blow-up in finite time, global attractors.
Mathematics Subject Classification:  Primary: 37L60, 34D45; Secondary: 35Q55.

Received: May 2006;      Revised: July 2007;      Available Online: September 2007.