Scattering above energy norm of solutions of a loglog energy-supercritical Schrodinger equation with radial data

Abstract

We prove scattering of $\tilde{H}^{k} $ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}} u \log^{c} {(\log{(10+|u|^{2})})}$, $0 < c < c_{n}$, $n={3,4}$, with radial data $u(0):=u_{0} \in \tilde{H}^{k} $, $k>n/2$. This is achieved, roughly speaking, by extending Bourgain's argument (see also Grillakis) and Tao's argument in high dimensions.