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Upper semicontinuity of attractors for the Klein-Gordon-Schrödinger equation. (English) Zbl 1067.35109

Summary: We consider the Klein Gordon-Schrödinger equation \[ i \frac {\partial \psi}{\partial t}+\Delta\psi+i\nu\psi+ \varphi\psi=f, \quad \frac {\partial^2 \varphi} {\partial t^2}+\gamma\frac{\partial\varphi} {\partial t}-\Delta\varphi+ \varphi-|\psi|^2=g, \] defined on \(\mathbb{R}^b\) \((n\leq 3)\) and \(\Omega_m=\{x\in \mathbb{R}^n:|x|\leq m\}\). Let \({\mathcal A}\) and \({\mathcal A}_m\) be the global attractors of the equation corresponding to \(\mathbb{R}^n\) and \(\Omega_m\), respectively. Then we prove that for any neighborhood \(U\) of \({\mathcal A}\), the global attractor \({\mathcal A}_m\) enters \(U\) when \(m\) is large enough.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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