The radial defocusing energy-supercritical NLS in dimension four

TL;DR

This paper proves that in four-dimensional space, all radial solutions to the defocusing nonlinear Schrödinger equation with supercritical nonlinearity that are bounded in the critical Sobolev space are globally well-behaved and scatter, extending understanding of supercritical NLS.

Contribution

It establishes global existence and scattering for radial solutions of supercritical defocusing NLS in four dimensions under boundedness in the critical Sobolev space.

Findings

Radial solutions with bounded critical Sobolev norm are global.

Such solutions scatter as time goes to infinity.

The result applies to supercritical exponents p>4 in 4D.

Abstract

We consider the radial defocusing nonlinear Schr\"odinger equations $iu_t+\Delta u=|u|^{p}u$ with supercritical exponent $p>4$ in four space dimensions, and prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter.