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

Abstract

We prove global well posedness and scattering for the nonlinear Schröodinger equation with power-type nonlinearity $$ \begin{cases} i u_t +\Delta u = |u|^p u, \quad \frac{4}{n} < p < \frac{4}{n-2},\\ u(0,x) = u_0(x)\in H^s({\mathbb R}^n), \quad n\geq 3, \end{cases} $$ below the energy space, i.e., for $s<1$. In [15], 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 a priori interaction Morawetz inequality to show that scattering holds in $H^s({\mathbb R}^n)$ whenever $s$ is larger than some value $0<s_0(n,p)<1$.