{"title":"On well-posedness for inhomogeneous Hartree equations in the critical case","authors":"Seongyeon Kim","doi":"10.3934/cpaa.2023060","DOIUrl":null,"url":null,"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.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/cpaa.2023060","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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。
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.