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.

A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

“…Our main technical tool is the theory of potential wells (see e.g. [5,7,8,12,13,17,21,24,26,27,29,31]) with a slight modification, which plays an essential role in the proofs of main results.…”

Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

“…However, we cannot expect the same for the nonlinear dynamical systems see [23], [22] and [21] as examples. Hence a lot of interest has been paid to the finite-time blow-up phenomena [26], [11], [4], not only for the parabolic type model [26], [21], [22], but also for the hyperbolic type model, and also the Schrödinger model [25]. Furthermore, combing the results from both sides of global existence and finite-time blow-up, we are also interested in the so-called sharp conditions [21], [23], [22].…”

On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term

“…(1.6), and then obtained a threshold value separating the existence and nonexistence of solutions. For the other types of second-order Schrödinger equation, many experts have paid attention to their qualitative properties and obtained abundant theoretical results, we refer the reader to see [30,19,2,36,32,7,6,1] and the papers cited therein.…”