Blow-up for the focusing H˙1/2-critical Hartree equation with radial data

In this paper, we consider the focusing H˙1/2-critical nonlinear Schrödinger equation (NLS) with Hartree type nonlinearityi∂tu+Δu=−(|⋅|−3⁎|u|2)uin R5. We prove that a solution, which blows up in kinetic energy in finite time, must blow up also in the H˙x1/2-norm. Moreover, we show that the Lx5/2-norm blows up with a lower bound‖u(t,⋅)‖Lx5/2⩾|log(T−t)|γas t→T, for some γ>0γ>0.