Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential

Abstract

The Cauchy problem of nonlinear Schrödinger equation with a harmonic potential for describing the attractive Bose-Einstein condensate under the magnetic trap is considered. We give some sufficient conditions of global existence and finite time blow up of solutions by introducing a family of potential wells. Some different sharp conditions for global existence, and some invariant sets of solutions are also obtained here.

Keywords

• nonlinear schrödinger equation
• harmonic potential
• potential wells
• global existence
• invariant sets