Well-posedness and blow-up properties for the generalized Hartree equation

We study the generalized Hartree equation, which is a nonlinear Schrodinger-type equation with a nonlocal potential iu(t) + Delta u + (vertical bar x vertical bar-b *vertical bar x vertical bar u vertical bar x vertical bar p)vertical bar x vertical bar u vertical bar x vertical bar(p-2)u = 0,x is an element of R-N. We establish the local well-posedness at the nonconserved critical regularity (H) over dot(sc) for s(c) >= 0, which also includes the energy-supercritical regime s(c) > 1 (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the H-1 well-posedness in the intercritical regime together with classification of solutions under the mass-energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass-energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.