Remarks on nonlinear Schrodinger equations with harmonic potential

Bose-Einstein condensation is usually modeled by nonlinear Schrodinger equations with harmonic potential. We study the Cauchy problem for these equations. We show that the local problem can be treated as in the case with no potential. For the global problem, we establish as evolution law, which is the analogue of the pseudo-conformal conversation law for the nonlinear Schrodinger equation. With this evolution law, we give wave collapse criteria, as well as an upper bound for the blow up time. Taking the physical scales into account, we finally give a lower bound for the breaking time. This study relies on two explicit operations, suited to nonlinear Schrodinger equations with harmonic potential, already known in the linear setting.