PROC
Global attractor for a Klein-Gordon-Schrodinger type system
Marilena N. Poulou Nikolaos M. Stavrakakis
Conference Publications 2007, 2007(Special): 844-854 doi: 10.3934/proc.2007.2007.844
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.
DCDS-S
Finite dimensionality of a Klein-Gordon-Schrödinger type system
Marilena N. Poulou Nikolaos M. Stavrakakis
Discrete & Continuous Dynamical Systems - S 2009, 2(1): 149-161 doi: 10.3934/dcdss.2009.2.149
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.

keywords: Klein-Gordon-Schrödinger System; Global Attractor; Absorbing Set; Lyapunov Exponents; Hausdorff and Fractal Dimension
DCDS-S
Global existence for a wave equation on $R^n$
Perikles G. Papadopoulos Nikolaos M. Stavrakakis
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 139-149 doi: 10.3934/dcdss.2008.1.139
We study the initial value problem for some degenerate non-linear dissipative wave equations of Kirchhoff type: $ u_{t t}-\phi (x)||\grad u(t)||^{2\gamma}\Delta u+\delta u_{t} = f(u),x\in R^n,t\geq 0,$ with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3, delta > 0, \gamma\geq 1$, $f(u)=|u|^{a}u$ with $a>0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(R^n)\cap L^{\infty}(R^n)$. If the initial data $\{ u_{0},u_{1}\}$ are small and $||\grad u_{0}||>0$, then the unique solution exists globally and has certain decay properties.
keywords: Quasilinear Hyperbolic Equations Generalised Sobolev Spaces Blow-Up Dissipation Global Solution Concavity Method Kirchhoff Strings Unbounded Domains Weighted $L^p$ Spaces. Potential Well

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