On well-posedness for inhomogeneous Hartree equations in the critical case

IF 1 3区 数学 Q1 MATHEMATICS Communications on Pure and Applied Analysis Pub Date : 2022-12-14 DOI:10.3934/cpaa.2023060
Seongyeon Kim
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引用次数: 3

Abstract

We study the well-posedness for the inhomogeneous Hartree equation $i\partial_t u + \Delta u = \lambda(I_\alpha \ast |\cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in $H^s$, $s\ge0$. Until recently, its well-posedness theory has been intensively studied, focusing on solving the problem for the critical index $p=1+\frac{2-2b+\alpha}{n-2s}$ with $0\le s \le 1$, but the case $1/2\leq s \leq 1$ is still an open problem. In this paper, we develop the well-posedness theory in this case, especially including the energy-critical case. To this end, we approach to the matter based on the Sobolev-Lorentz space which can lead us to perform a finer analysis for this equation. This is because it makes it possible to control the nonlinearity involving the singularity $|x|^{-b}$ as well as the Riesz potential $I_\alpha$ more effectively.
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临界情况下非齐次Hartree方程的适定性
在$H^s$, $s\ge0$中研究了非齐次Hartree方程$i\partial_t u + \Delta u = \lambda(I_\alpha \ast |\cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$的适定性。直到最近,人们对其适定性理论进行了深入的研究,主要集中在用$0\le s \le 1$求解临界指标$p=1+\frac{2-2b+\alpha}{n-2s}$的问题上,但情况$1/2\leq s \leq 1$仍然是一个悬而未决的问题。在本文中,我们发展了这种情况下的适定性理论,特别是包括能量临界情况。为此,我们基于索博列夫-洛伦兹空间来处理这个问题,这可以使我们对这个方程进行更精细的分析。这是因为它可以更有效地控制涉及奇点$|x|^{-b}$以及Riesz势$I_\alpha$的非线性。
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来源期刊
CiteScore
1.90
自引率
10.00%
发文量
124
审稿时长
4-8 weeks
期刊介绍: CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. It is edited by a group of energetic leaders to guarantee the journal''s highest standard and closest link to the scientific communities.
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