Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
121 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On well-posedness and blow-up in the generalized Hartree equation (1910.01085v1)

Published 2 Oct 2019 in math.AP

Abstract: We study the generalized Hartree equation, which is a nonlinear Schr\"odinger-type equation with a nonlocal potential $iu_t + \Delta u + (|x|{-b} \ast |u|p)|u|{p-2}u=0, x \in \mathbb{R}N$.We establish the local well-posedness at the non-conserved critical regularity $\dot{H}{s_c}$ for $s_c \geq 0$, which also includes the energy-supercritical regime $s_c>1$ (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.

Summary

We haven't generated a summary for this paper yet.