Sharp Criteria of Scattering for the Fractional NLS

Abstract

In this paper, the sharp threshold of scattering for the fractional nonlinear Schrödinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\frac{4s}{N}<p<1+\frac{4s}{N-2s}$, and $$M[u_0{]}^{\frac{s-s_c}{s_c}}E[u_0]<M[Q{]}^{\frac{s-s_c}{s_c}}E[Q],M[u_0{]}^{\frac{s-s_c}{s_c}}\Vert u_0{\Vert }_{{\dot{H}}^s}^2<M[Q{]}^{\frac{s-s_c}{s_c}}\Vert Q{\Vert }_{{\dot{H}}^s}^2$$ then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that if $1+\frac{4s}{N}<p<1+\frac{4s}{N-2s}$ and $$M[u_0{]}^{\frac{s-s_c}{s_c}}E[u_0]<M[Q{]}^{\frac{s-s_c}{s_c}}E[Q],M[u_0{]}^{\frac{s-s_c}{s_c}}\Vert u_0{\Vert }_{{\dot{H}}^s}^2>M[Q{]}^{\frac{s-s_c}{s_c}}\Vert Q{\Vert }_{{\dot{H}}^s}^2,$$ then the corresponding solution $u(t)$ blows up in finite time, according to Boulenger, Himmelsbach, and Lenzmann's results in [2].

arXiv admin note: text overlap with arXiv:1705.08615

References in corpus (2)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Nonlinear fractional Schrödinger equations in one dimension