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

Abstract

In this paper, we consider the defocusing nonlinear Schrödinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is $priori$ 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 practice, we use weighted Strichartz space adapted for our setting which ultimately helps us solve the problems in cases $d\geq 4$ and $0