On well-posedness and blow-up in the generalized Hartree equation

Abstract

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential $i{u}_t+\Delta u+(|x{|}^{-b}\ast |u{|}^p)|u{|}^{p-2}u=0,x\in {\mathbb{R}}^N$.We establish the local well-posedness at the non-conserved critical regularity ${\dot{H}}^{s_c}$ for $s_c\ge 0$, which also includes the energy-supercritical regime $s_c>1$ (thus, complementing the work in [3], where the authors 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). Both of these results hold regardless of the criticality of the equation. 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.

References in corpus (3)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
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• Collapse and stable self-trapping for Bose-Einstein condensates with 1/r^b type attractive interatomic interaction potential

Cited by in corpus (1)

• Stable blow-up dynamics in the $L^2$-critical and $L^2$-supercritical generalized Hartree equation