We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{ 0}]^{ \frac{ s-s_c}{ s_c}}E[u_{ 0}<M[Q]^{ \frac{ s-s_c}{ s_c}}E[Q]$ and $M[u_{ 0}]^{ \frac{ s-s_c}{ s_c}}\|u_{ 0}\|^2_{ \dot H^s}<M[Q]^{ \frac{ s-s_c}{ s_c}}\| Q\|^2_{ \dot H^s}$, then the solution $u(t)$ is globally well-posed and scatters; if $ M[u_{ 0}]^{ \frac{ s-s_c}{ s_c}}E[u_{ 0}]<M[Q]^{ \frac{ s-s_c}{ s_c}}E[Q]$ and $M[u_{ 0}]^{ \frac{ s-s_c}{ s_c}}\|u_{ 0}\|^2_{ \dot H^s}>M[Q]^{ \frac{ s-s_c}{ s_c}}\| Q\|^2_{ \dot H^s}$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{ it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, $Q$ is the ground-state solution for the fractional Hartree equation.
Categories: math.AP