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Finite dimensionality of a Klein-Gordon-Schrödinger type system

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  • 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.

    Mathematics Subject Classification: Primary: 35B40, 35B45, 35B65, 35D05, 35D10, 35J50 ; Secondary: 35J70, 35P30.

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