Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
131 tokens/sec
GPT-4o
10 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

The energy-critical inhomogeneous generalized Hartree equation in 3D (2305.00972v2)

Published 1 May 2023 in math.AP

Abstract: The purpose of this work is to study the $3D$ energy-critical inhomogeneous generalized Hartree equation $$ i\pa_tu+\Delta u+|x|{-b}(I_\alpha\ast|\cdot|{-b}|u|{p})|u|{p-2}u=0,\;\ x\in\R3, $$ where $p=3+\alpha-2b$. We establish global well-posedness and scattering below the ground state threshold with non-radial initial data in $\dot{H}1$. To this end, we exploit the decay of the nonlinearity, which together with the Kenig-Merle roadmap, allows us to treat the non-radial case as the radial case. In this paper are introduced new techniques to overcome the challenges posed by the presence of the potential and the nonlocal nonlinear term of convolution type. In particular, we also show scattering for the classical generalized Hartree equation ($b=0$) assuming radial data. Additionally, in the defocusing case, we show scattering with general data. We believe that the ideas developed here are robust and can be applicable to other types of nonlinear Hartree equations. In the introduction, we discuss some open problems.

Summary

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