Abstract
In this paper, we study the Cauchy problem for nonlinear Schrödinger equations arising as an effective model of the Bose–Einstein condensate in a magnetic trap rotating with an angular velocity. We first establish sufficient conditions showing the existence of global-in-time and finite time blow-up solutions to the equation. We next derive sharp thresholds for global existence versus finite time blow-up in the mass-critical and mass-supercritical cases. We also study the existence and strong instability of ground state standing waves related to the equation. Finally we prove the existence, non-existence, and orbital stability of prescribed mass standing waves when the rotational speed is smaller than a critical value.
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