Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

We consider a nonlinear Schrödinger equation in a time-dependent domain Qτ of ℝ2 given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{\tau} - i u_{\varepsilon\varepsilon} + |u|^{2} u + \gamma v=0. $$\end{document} We prove the well-posedness of the above model and analyze the behaviour of the solution as t→+∞. We consider two situations: the conservative case (γ=0) and the dissipative case (γ>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.