2019/10/02 by Anudeep Kumar Arora, Svetlana Roudenko, Arora, Anudeep K. +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1910.01085
openalex publication_date 2019/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential iut + Δu + (|x|-b ∗ |u|p)|u|p-2u=0, x ∈ ℝN.We establish the local well-posedness at the non-conserved critical regularity Hsc for sc ≥ 0, which also includes the energy-supercritical regime sc>1 (thus, complementing the work in [3], where the authors obtained the H1 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.