Large data mass-subcritical NLS: critical weighted bounds imply scattering

TL;DR

This paper proves that for mass-subcritical nonlinear Schrödinger equations, solutions with certain weighted bounds are guaranteed to be global and scatter, extending understanding of solution behavior in all space dimensions.

Contribution

It establishes that critical weighted bounds imply global existence and scattering for all solutions in the mass-subcritical NLS with various nonlinearities.

Findings

Weighted bounds ensure global solutions.

Solutions with critical regularity scatter.

Results apply to all space dimensions.

Abstract

We consider the mass-subcritical nonlinear Schr\"odinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution satisfying $\|\, |x|^{|s_c|}e^{-it\Delta} u\|_{L_t^\infty L_x^2} <\infty$ on its maximal interval of existence must be global and scatter.