Global well-posedness for the three-dimensional |u|^2 quadratic NLS

Establish global well-posedness for the three-dimensional quadratic nonlinear Schrödinger equation $(i\\partial_t-\\Delta)u=|u|^2$; the problem remains unresolved even for localized and smooth initial data when using the space-time resonance method.

Background

The paper analyzes the quadratic nonlinear Schrödinger equation in three spatial dimensions, principally focusing on the $u^2$ component because its resonant structure can be treated using trilinear restriction estimates and a new transference method. The authors explicitly distinguish the $|u|^2$ component as substantially more difficult: although local well-posedness and almost-global existence results are known in certain settings, the high–high-to-low interaction lacks the modulation separation available for $u^2$.

The unresolved problem concerns global-in-time well-posedness for the $|u|^2$ nonlinearity, even under the favorable assumptions of smoothness and spatial localization, particularly through the space-time resonance framework.