The Long-Time Behavior of Spectral Approximate for Klein-Gordon-Schrödinger Equations

Abstract

Klein-Gordon-Schrödinger (KGS) equations are very important in physics. Some papers studied their well-posedness and numerical solution [1-4], and another works investigated the existence of global attractor in $R^n$ and $\Omega ⊂ R^n \ (n\leq 3)$ [5-6,11-12]. In this paper, we discuss the dynamical behavior when we apply spectral method to find numerical approximation for periodic initial value problem of KGS equations. It includes the existence of approximate attractor $A_N$, the upper semi-continuity on $A$ which is a global attractor of initial problem and the upper bounds of Hausdorff and fractal dimensions for $A$ and $A_N$, etc.