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

We consider a nonlinear Schrödinger equation in a time-dependent domain Q τ of ℝ 2 given by 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.