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.