The focusing energy-critical Hartree equation

We consider the focusing energy-critical nonlinear Hartree equation iut+Δu=−(⁻⁴|x|∗²|u|)u. We proved that if a maximal-lifespan solution u:I×Rd→C satisfies supt∈I‖∇u(t)_‖2<‖∇W_‖2, where W is the static solution of the equation, then the maximal-lifespan I=R, moreover, the solution scatters in both time directions. For spherically symmetric initial data, similar result has been obtained in C. Miao, G. Xu, L. Zhao, Global wellposedness, scattering and blowup for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., in press]. The argument is an adaptation of the recent work of R. Killip and M. Visan R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrödinger equations.