{"paper":{"title":"Scattering for the focusing ${\\dot H}^{1/2}$-critical Hartree equation with radial data","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Yanfang Gao","submitted_at":"2009-06-18T09:00:23Z","abstract_excerpt":"We investigate the focusing $\\dot H^{1/2}$-critical nonlinear Schr\\\"{o}dinger equation (NLS) of Hartree type $i\\partial_t u + \\Delta u = -(|\\cdot|^{-3} \\ast |u|^2)u$ with $\\dot H^{1/2}$ radial data in dimension $d = 5$. It is proved that if the maximal life-span solution obeys $\\sup_{t}\\big\\||\\nabla|^{{1/2}}u\\big\\|_2 < \\frac{\\sqrt{6}}{3} \\big\\||\\nabla|^{{1/2}}Q\\big\\|_2$, where $Q$ is the positive radial solution to the elliptic equation(\\ref{e14}). Then the solution is global and scatters."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.3382","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0906.3382/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}