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Global attractor for a Klein-Gordon-Schrodinger type system

Pages: 844 - 854, Issue Special, September 2007

 Abstract        Full Text (208.5K)              

Marilena N. Poulou - Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Greece (email)
Nikolaos M. Stavrakakis - Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Hellas, Greece (email)

Abstract: In this paper we prove the existence and uniqueness of solutions for the following evolution system of Klein-Gordon-Schrodinger type

$i\psi_t + k\psi_(xx) + i\alpha\psi$ = $\phi\psi + f(x)$,

$\phi_(tt)$ - $\phi_(xx) + \phi + \lambda\phi_t$ = -$Re\psi_x + g(x)$,

$\psi(x,0)=\psi_0(x), \phi(x,0)$ = $\phi_0, \phi_t(x,0)=\phi_1(x)$

$\phi(x,t)=\phi(x,t)=0$, $x\in\partial\Omega, t>0$

where $x \in \Omega, t > 0, k > 0, \alpha > 0, \lambda > 0, f(x)$ and $g(x)$ are the driving terms and $\Omega$ (bounded) $\subset \mathbb{R}$. Also we prove the continuous dependence of solutions of the system on the initial data as well as the existence of a global attractor.

Keywords:  Klein-Gordon-Schrodinger equation; Global Attractor; Absorbing set; Asymptotic Compactness; Uniqueness; Continuity.
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35.

Received: September 2006;      Revised: July 2007;      Published: September 2007.