On the focusing energy-critical fractional nonlinear schrödnger equations

We consider the fractional nonlinear Schrödinger equation (FNLS) with non-local dispersion |∇|^α and focusing energy-critical Hartree type nonlinearity [-(|x|^-2α * |u|²)u]. We first establish a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments [20] when the initial energy and initial kinetic energy are less than those of ground state, respectively. We revisit and highlight long time perturbation, profile decomposition and localized virial inequality. As an application of the localized virial inequality, we provide a proof for finite time blowup for energy critical Hartree equations via commutator technique introduced in [2].