Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system.
The authors consider the coupled Klein-Gordon-Schrödinger system \[ iu_t+\Delta u+i\nu u+\phi u=f, \] \[ \phi_{tt}+\gamma \phi_t-\Delta\phi+\phi-| u| ^2=g, \] for \((u,\phi):\mathbb R\times\Omega\to\mathbb C\times\mathbb R\), where \(\Omega\subset\mathbb R^n\) is smooth and bounded or all of~\(\mathbb R^n\), \(n\leq3\), with Dirichlet boundary conditions, \(\gamma,\nu\) are positive constants, and \(f,g\) belong to \(L^2\). The system is considered as an evolution equation with initial condition \((u,\phi,\phi_t)\). It is known that the system has an attractor~\(A_1\) for the induced semigroup on \(E_1=H_0^1(\Omega)\times H_0^1(\Omega)\times L^2(\Omega)\), and also that it has an attractor on \(E_2=\bigl(H_0^1(\Omega)\cap H^2(\Omega)\bigr) \times\bigr(H_0^1(\Omega)\cap H^2(\Omega)\bigr) \times H_0^1(\Omega)\). The main result states that~\(A_1\) is a compact subset of~\(E_2\) already, which can be interpreted as an asymptotical smoothing effect.
scientific article; zbMATH DE number 2004913