---
OA_place: repository
OA_type: green
_id: '22023'
abstract:
- lang: eng
  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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  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>
  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>
  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>.
  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.
  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.
  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.
date_created: 2026-06-19T07:31:39Z
date_published: 2010-03-31T00:00:00Z
date_updated: 2026-06-19T10:24:15Z
day: '31'
doi: 10.1353/ajm.0.0107
extern: '1'
external_id:
  arxiv:
  - '0804.1018'
intvolume: '       132'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.0804.1018
month: '03'
oa: 1
oa_version: Preprint
page: 361-424
publication: American Journal of Mathematics
publication_identifier:
  eissn:
  - 1080-6377
  issn:
  - 0002-9327
publication_status: published
publisher: Johns Hopkins University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: The focusing energy-critical nonlinear Schrödinger equation in dimensions five
  and higher
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2010'
...