Global well-posedness, scattering, and blowup for nonlinear coupled Schrödinger equations in ℝ³

Abstract

In this paper, we study the system of the coupled nonlinear Schröd-inger equation in dimension three. By establishing the a priori interaction, Morawetz estimate in the spirit of the work of Tao et al. [The nonlinear Schrödinger equation with combined power-type nonlinearities. Commun. Par. Differ. Equ. 2007;32:1281–1343], we prove the global well-posedness and scattering of the solution to the defocusing coupled equation. Furthermore, we adapt the standard method from Cazenave [Semilinear Schrödinger equations. Vol. 10, Courant lecture notes in mathematics. Providence (RI): American Mathematical Society; 2003] to obtain the blowup result for the focusing case under the restriction: energy is negative and the initial data belongs to or the initial data is radial.

Keywords:

• nonlinear Schrödinger combined equations
• well-posedness
• interaction Morawetz estimate
• blow-up

AMS Subject Classifications:

• 35P25
• 35Q55