Global well-posedness and scattering for a class of nonlinear Schrodinger equations below the energy space

Abstract

We prove global well-posedness and scattering for the nonlinear Schrödinger equation with power-type nonlinearity \begin{equation*} \begin{cases} i u_t +Δu = |u|^p u, \quad \frac{4}{n}<p<\frac{4}{n-2}, u(0,x) = u_0(x)\in H^s(\R^n), \quad n\geq 3, \end{cases} \end{equation*} below the energy space, i.e., for $s<1$. In \cite{ckstt:low7}, J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao established polynomial growth of the $H^s_x$-norm of the solution, and hence global well-posedness for initial data in $H^s_x$, provided $s$ is sufficiently close to 1. However, their bounds are insufficient to yield scattering. In this paper, we use the \emph{a priori} interaction Morawetz inequality to show that scattering holds in $H^s(\R^n)$ whenever $s$ is larger than some value $0<s_0(n,p)<1$.