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

We prove scattering of ${\tilde{H}}^k$ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}} u \log^{c} {(\log{(10+|u|^{2})})}$,$0 < c < c_{n}$,$n={3,4}$,withradialdata$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.