The interctitical defocusing nonlinear Schr\"odinger equations with radial initial data in dimensions four and higher

TL;DR

This paper proves that radial solutions to the defocusing nonlinear Schrödinger equation in dimensions four and higher, which are bounded in the critical Sobolev space, are global and scatter, extending previous results to higher dimensions.

Contribution

It extends the global well-posedness and scattering results for radial solutions of the defocusing NLS to dimensions four and above for solutions bounded in the critical Sobolev space.

Findings

Radial solutions are global and scatter under certain conditions.

Weighted Strichartz spaces are effective in higher dimensions.

Extension of previous results to dimensions d ≥ 4.

Abstract

In this paper, we consider the defocusing nonlinear Schr\"odinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is \emph{priori} bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot{H}^{s_c}_x$, then $u$ is global and scatters. In practise, we use weighted Strichartz space adapted for our setting which ultimately helps us solve the problems in cases $d\geq 4$ and $0<s_c<\f{1}{2}$. The results in this paper extend the work of \cite[Comm. in PDEs, 40(2015), 265-308]{M3} to higher dimensions.