{"title":"Scattering for the generalized Hartree equation with a potential","authors":"Carlos M. Guzmán, Cristian Loli, Luis P. Yapu","doi":"arxiv-2409.10769","DOIUrl":null,"url":null,"abstract":"We consider the focusing generalized Hartree equation in $H^1(\\R^3)$ with a\npotential, \\begin{equation*} iu_t + \\Delta u - V(x)u + (I_\\gamma \\ast |u|^p\n)|u|^{p-2} u=0, \\end{equation*} where $I_\\gamma = \\frac{1}{|x|^{3-\\gamma}}$, $p\n\\geq 2$ and $\\gamma < 3$. In this paper, we prove scattering for the\ngeneralized Hartree equation with a potential in the intercritical case\nassuming radial initial data. The novelty of our approach lies in the use of a\ngeneral mass-potential condition, incorporating the potential V, which extends\nthe standard mass-energy framework. To this end, we employ a simplified method\ninspired by Dodson and Murphy \\cite{Dod-Mur}, based on Tao's scattering\ncriteria and Morawetz estimates. This approach provides a more straightforward\nproof of scattering compared to the traditional\nconcentration-compactness/rigidity method of Kenig and Merle \\cite{KENIG}.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"23 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10769","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the focusing generalized Hartree equation in $H^1(\R^3)$ with a
potential, \begin{equation*} iu_t + \Delta u - V(x)u + (I_\gamma \ast |u|^p
)|u|^{p-2} u=0, \end{equation*} where $I_\gamma = \frac{1}{|x|^{3-\gamma}}$, $p
\geq 2$ and $\gamma < 3$. In this paper, we prove scattering for the
generalized Hartree equation with a potential in the intercritical case
assuming radial initial data. The novelty of our approach lies in the use of a
general mass-potential condition, incorporating the potential V, which extends
the standard mass-energy framework. To this end, we employ a simplified method
inspired by Dodson and Murphy \cite{Dod-Mur}, based on Tao's scattering
criteria and Morawetz estimates. This approach provides a more straightforward
proof of scattering compared to the traditional
concentration-compactness/rigidity method of Kenig and Merle \cite{KENIG}.