The focusing energy-critical Hartree equation

Abstract

We consider the focusing energy-critical nonlinear Hartree equation i u t + Δ u = − ( | x | −4 ∗ | u | 2 ) u . We proved that if a maximal-lifespan solution u : I × R d → C satisfies sup t ∈ 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.