Asymptotic behavior of the solutions of a system of Schödinger-Poisson equations on a bounded domain in \(\mathbb R ^3 \)
The weakly damped driven Schrödinger-Poisson system \[ \partial_t u+\gamma u+i\Delta u+iu\varphi=f , \qquad \pm\Delta\varphi=|u|^2 \] for \((t,x)\in (0,\infty)\times\Omega\) is considered in a bounded domain \(\Omega\) of \(\mathbb R^3\), supplemented with homogeneous Dirichlet boundary conditions for \(u\) and \(\varphi\) and an initial condition \(u_0\in H^1_0(\Omega)\). Assuming that the driving force \(f\) belongs to \(L^2(\Omega)\) and the damping parameter \(\gamma\) is positive, the existence of a compact attractor with finite Hausdorff dimension in \(H_0^1(\Omega)\) is shown.
homogeneous Dirichlet boundary conditions
scientific article; zbMATH DE number 5761667