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2009, 2(1): 149-161. doi: 10.3934/dcdss.2009.2.149

Finite dimensionality of a Klein-Gordon-Schrödinger type system

1. 

Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Greece

2. 

Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Hellas, Greece

Received  February 2008 Revised  October 2008 Published  January 2009

In this paper we study the finite dimensionality of the global attractor for the following system of Klein-Gordon-Schrödinger type

$ i\psi_t +\kappa \psi_{xx} +i\alpha\psi = \phi\psi+f,$
$ \phi_{tt}- \phi_{xx}+\phi+\lambda\phi_t = -Re \psi_{x}+g, $
$\psi (x,0)=\psi_0 (x), \phi(x,0) = \phi_0 (x), \phi_t (x,0)=\phi_1(x),$
$ \psi(x,t)= \phi(x,t)=0, x \in \partial \Omega, t>0, $

where $x \in \Omega, t>0, \kappa > 0, \alpha >0, \lambda >0,$ $f$ and $g$ are driving terms and $\Omega$ is a bounded interval of R With the help of the Lyapunov exponents we give an estimate of the upper bound of its Hausdorff and Fractal dimension.

Citation: Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149
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