{"title":"Regular Strichartz estimates in Lorentz-type spaces with application to the $H^s$-critical inhomogeneous biharmonic NLS equation","authors":"RoeSong Jang, JinMyong An, JinMyong Kim","doi":"arxiv-2409.06278","DOIUrl":null,"url":null,"abstract":"In this paper, we investigate the Cauchy problem for the $H^s$-critical\ninhomogeneous biharmonic nonlinear Schr\\\"{o}dinger (IBNLS) equation \\[iu_{t}\\pm\n\\Delta^{2} u=\\lambda |x|^{-b}|u|^{\\sigma}u,~u(0)=u_{0} \\in H^{s} (\\mathbb\nR^{d}),\\] where $\\lambda\\in \\mathbb C$, $d\\ge 3$, $1\\le s<\\frac{d}{2}$,\n$0<b<\\min \\left\\{4,2+\\frac{d}{2}-s \\right\\}$ and $\\sigma=\\frac{8-2b}{d-2s}$.\nFirst, we study the properties of Lorentz-type spaces such as Besov-Lorentz\nspaces and Triebel-Lizorkin-Lorentz spaces. We then derive the regular\nStrichartz estimates for the corresponding linear equation in Lorentz-type\nspaces. Using these estimates, we establish the local well-posedness as well as\nthe small data global well-posedness and scattering in $H^s$ for the\n$H^s$-critical IBNLS equation under less regularity assumption on the nonlinear\nterm than in the recent work \\cite{AKR24}. This result also extends the ones of\n\\cite{SP23,SG24} by extending the validity of $d$, $b$ and $s$. Finally, we\ngive the well-posedness result in the homogeneous Sobolev spaces $\\dot{H}^s$.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"109 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","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-2409.06278","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
In this paper, we investigate the Cauchy problem for the $H^s$-critical
inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t}\pm
\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb
R^{d}),\] where $\lambda\in \mathbb C$, $d\ge 3$, $1\le s<\frac{d}{2}$,
$0
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