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December  2007, 19(4): 711-736. doi: 10.3934/dcds.2007.19.711

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

N. I. Karachalios 1, , Hector E. Nistazakis 2, and  Athanasios N. Yannacopoulos 3,

1. 

Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
2. 

Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi GR 83200, Samos, Greece
3. 

Department of Statistics, Athens University of Economics and Business, Patission 76 GR 10434, Athens, Greece

Received  May 2006 Revised  July 2007 Published  September 2007

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: global attractors., lattice differential equations, blow-up in finite time, Discrete Ginzburg-Landau equation.
Mathematics Subject Classification: Primary: 37L60, 34D45; Secondary: 35Q5.
Citation: N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
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