[{"article_type":"original","date_published":"2010-03-31T00:00:00Z","issue":"2","title":"The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher","date_created":"2026-06-19T07:31:39Z","author":[{"first_name":"Rowan","full_name":"Killip, Rowan","last_name":"Killip"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan","full_name":"Visan, Monica","first_name":"Monica"}],"intvolume":"       132","doi":"10.1353/ajm.0.0107","volume":132,"scopus_import":"1","year":"2010","publication":"American Journal of Mathematics","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.0804.1018"}],"status":"public","day":"31","publication_identifier":{"issn":["0002-9327"],"eissn":["1080-6377"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","publication_status":"published","extern":"1","external_id":{"arxiv":["0804.1018"]},"OA_type":"green","abstract":[{"text":"We consider the focusing energy-critical nonlinear Schrödinger equation iut + ∆u = −|u|\r\n4 d−2 u in dimensions d ≥ 5. We prove that if a maximal-lifespan solution u : I × Rd → C obeys supt∈I k∇u(t)k2 < k∇Wk2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In\r\nparticular, if a solution has both energy and kinetic energy less than those\r\nof the ground state W at some point in time, then the solution is global and\r\nscatters. We also show that any solution that blows up with bounded kinetic\r\nenergy must concentrate at least the kinetic energy of the ground state. Similar\r\nresults were obtained by Kenig and Merle in [17, 18] for spherically symmetric\r\ninitial data and dimensions d = 3, 4, 5.","lang":"eng"}],"_id":"22023","month":"03","citation":{"mla":"Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>, vol. 132, no. 2, Johns Hopkins University Press, 2010, pp. 361–424, doi:<a href=\"https://doi.org/10.1353/ajm.0.0107\">10.1353/ajm.0.0107</a>.","ieee":"R. Killip and M. Vişan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,” <i>American Journal of Mathematics</i>, vol. 132, no. 2. Johns Hopkins University Press, pp. 361–424, 2010.","apa":"Killip, R., &#38; Vişan, M. (2010). The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. <i>American Journal of Mathematics</i>. Johns Hopkins University Press. <a href=\"https://doi.org/10.1353/ajm.0.0107\">https://doi.org/10.1353/ajm.0.0107</a>","short":"R. Killip, M. Vişan, American Journal of Mathematics 132 (2010) 361–424.","ama":"Killip R, Vişan M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. <i>American Journal of Mathematics</i>. 2010;132(2):361-424. doi:<a href=\"https://doi.org/10.1353/ajm.0.0107\">10.1353/ajm.0.0107</a>","ista":"Killip R, Vişan M. 2010. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. 132(2), 361–424.","chicago":"Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>. Johns Hopkins University Press, 2010. <a href=\"https://doi.org/10.1353/ajm.0.0107\">https://doi.org/10.1353/ajm.0.0107</a>."},"type":"journal_article","page":"361-424","oa":1,"arxiv":1,"OA_place":"repository","oa_version":"Preprint","publisher":"Johns Hopkins University Press","quality_controlled":"1","date_updated":"2026-06-19T10:24:15Z","language":[{"iso":"eng"}]}]