The focusing energy-critical Hartree equation

We consider the focusing energy-critical nonlinear Hartree equation iu(t) + Delta u = -(vertical bar x vertical bar(-4) * vertical bar u vertical bar(2))u. We proved that ifa maximal-lifespan solution u : I x R-d -> C Satisfies sup(t is an element of I) parallel to del u(t)parallel to(2) < parallel to del W parallel to(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,in adaptation of the recent work of R. Killip, and M. Visan [R. Killip, M. Visan, The focusing energy-critical nonlinear Schodinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrodinger equations. (C) 2008 Elsevier Inc. All rights reserved.