Abstract
This note studies some asymptotic properties of global solutions to the non-linear fractional Schrodinger equation
<(i)over dot><(u)over dot> - (-Delta)(s)u = F(u), 0 < s < 1.
In the energy-sub-critical attractive regime, for both cases, a monomial or non-local source term, one proves that some Lebesgue norms of the energy global solutions, with spherically symmetric datum, vanish for large time. To avoid a loss of regularity in Strichartz estimates, one considers the spherically symmetric regime. The proof is based on a Morawetz estimate coupled with Strichartz inequalities, following the method of Visciglia (Math Res Lett 16:919-926, 2009). As a consequence, one revisits the scattering in the inter-critical regime.