The Interctitical Defocusing Nonlinear Schrödinger Equations with Radial Initial Data in Dimensions Four and Higher

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.