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 ℝ² given by u_τ- i u_{ε ε}} + |u|² u + γ v=0. 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.