Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential

Abstract

This paper is concerned with the nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose-Einstein condensate under the magnetic trap. By combining the best constant of Gagliardo-Nirenberg's inequality with the characteristic of this equation, we derive out a global existence condition for the supercritical equation which coincides with the critical case.

Sharp constant of an improved Gagliardo–Nirenberg inequality and its application

“…3) in the case of supercritical nonlinearity 1 + (4 + 2b)/N < p <p. An interesting aspect is that we can get the global solutions for arbitrarily large data. We will use Theorem 1.1 and some ideas from [2,20], which was used in [6]. We emphasize that the use of Theorem 1.1 is essential.…”

“…Remark 1.2 When K (x) = 1, Fukuizumi [9] , Rose and Weinstein [14] and Zhang [6,23] obtained some results of the stability and instability of standing waves as well as global existence of (1.2) for various initial profiles. It is observed that there is a basic estimate used in their papers, i.e.…”

Energy criteria of global existence for the coupled nonlinear Schrödinger equations with harmonic potentials

“…And under what conditions, will the condensates exist for all time (global existence)? Especially the sharp criteria of blowing up and global existence are pursued strongly(also see [7,8,9,23,25,26]). From the view point of mathematics, the problems are rather fascinating and challenging.…”

Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential

“…Chen and Zhang [13] derived a global existence condition for the supercritical case for (3), which coincides with the critical case. For (4), Fujiwara [14] proved that the smoothness of the Schrödinger kernel for potentials of quadratic growth.…”