{"publication":"American Journal of Mathematics","publication_identifier":{"eissn":["1080-6377"]},"article_processing_charge":"No","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.0804.1018","open_access":"1"}],"date_created":"2026-06-19T08:48:46Z","status":"public","oa_version":"Preprint","intvolume":"       132","date_updated":"2026-07-01T12:52:05Z","abstract":[{"text":"We consider the focusing energy-critical nonlinear Schr\\\"odinger equation \r\n$iu_t+\\Delta u = - |u|^{4\\over{d-2}}u$ in dimensions $d\\geq 5$. We prove \r\nthat if a maximal-lifespan solution $u\\colon \\ I\\times {\\Bbb R}^d\\to {\\Bbb C}$ obeys $\\sup_{t\\in I}\\|\\nabla u(t)\\|_2&lt;\\|\\nabla W\\|_2$, then it is \r\nglobal and scatters both forward and backward in time. Here $W$ denotes \r\nthe ground state, which is a stationary solution of the equation. In \r\nparticular, if a solution has both energy and kinetic energy less than \r\nthose of the ground state $W$ at some point in time, then the solution is \r\nglobal and scatters. We also show that any solution that blows up with \r\nbounded kinetic energy must concentrate at least the kinetic energy of the \r\nground state. Similar results were obtained by Kenig and Merle for \r\nspherically symmetric initial data and dimensions $d=3,4,5$.\r\n</jats:p>","lang":"eng"}],"author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica","last_name":"Visan"}],"OA_type":"green","title":"The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher","day":"01","quality_controlled":"1","year":"2010","external_id":{"arxiv":["0804.1018"]},"das_tickbox":"1","page":"361-424","article_type":"original","extern":"1","oa":1,"language":[{"iso":"eng"}],"doi":"10.1353/ajm.0.0107","publisher":"Johns Hopkins University Press","date_published":"2010-04-01T00:00:00Z","issue":"2","publication_status":"published","citation":{"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>","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>.","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>.","short":"R. Killip, M. Vişan, American Journal of Mathematics 132 (2010) 361–424."},"type":"journal_article","volume":132,"OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"22089","arxiv":1,"scopus_import":"1","month":"04"}