---
OA_place: publisher
OA_type: free access
_id: '22022'
abstract:
- lang: eng
  text: "We consider both the defocusing and focusing cubic nonlinear Klein–Gordon
    equations\r\nutt − Δu + u ± u3 =0\r\nin two space dimensions for real-valued initial
    data u(0) ∈ H1/x and ut(0) ∈ L2/x.\r\nWe show that in the defocusing case, solutions
    are global and have finite global L4/t,x spacetime bounds. In the focusing case,
    we characterize the dichotomy\r\nbetween this behaviour and blowup for initial
    data with energy less than that\r\nof the ground state.\r\nThese results rely
    on analogous statements for the two-dimensional cubic\r\nnonlinear Schr¨odinger
    equation, which are known in the defocusing case and\r\nfor spherically-symmetric
    initial data in the focusing case. Thus, our results\r\nare mostly unconditional.\r\nIt
    was previously shown by Nakanishi that spacetime bounds for Klein–\r\nGordon equations
    imply the same for nonlinear Schr¨odinger equations."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Betsy
  full_name: Stovall, Betsy
  last_name: Stovall
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Killip R, Stovall B, Vişan M. Scattering for the cubic Klein-Gordon equation
    in two space dimensions. <i>Transactions of the American Mathematical Society</i>.
    2012;364(3):1571-1631. doi:<a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">10.1090/s0002-9947-2011-05536-4</a>
  apa: Killip, R., Stovall, B., &#38; Vişan, M. (2012). Scattering for the cubic Klein-Gordon
    equation in two space dimensions. <i>Transactions of the American Mathematical
    Society</i>. American Mathematical Society. <a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">https://doi.org/10.1090/s0002-9947-2011-05536-4</a>
  chicago: Killip, Rowan, Betsy Stovall, and Monica Vişan. “Scattering for the Cubic
    Klein-Gordon Equation in Two Space Dimensions.” <i>Transactions of the American
    Mathematical Society</i>. American Mathematical Society, 2012. <a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">https://doi.org/10.1090/s0002-9947-2011-05536-4</a>.
  ieee: R. Killip, B. Stovall, and M. Vişan, “Scattering for the cubic Klein-Gordon
    equation in two space dimensions,” <i>Transactions of the American Mathematical
    Society</i>, vol. 364, no. 3. American Mathematical Society, pp. 1571–1631, 2012.
  ista: Killip R, Stovall B, Vişan M. 2012. Scattering for the cubic Klein-Gordon
    equation in two space dimensions. Transactions of the American Mathematical Society.
    364(3), 1571–1631.
  mla: Killip, Rowan, et al. “Scattering for the Cubic Klein-Gordon Equation in Two
    Space Dimensions.” <i>Transactions of the American Mathematical Society</i>, vol.
    364, no. 3, American Mathematical Society, 2012, pp. 1571–631, doi:<a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">10.1090/s0002-9947-2011-05536-4</a>.
  short: R. Killip, B. Stovall, M. Vişan, Transactions of the American Mathematical
    Society 364 (2012) 1571–1631.
das_tickbox: '1'
date_created: 2026-06-19T07:31:10Z
date_published: 2012-03-01T00:00:00Z
date_updated: 2026-06-22T09:19:43Z
day: '01'
doi: 10.1090/s0002-9947-2011-05536-4
extern: '1'
external_id:
  arxiv:
  - '1008.2712'
intvolume: '       364'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1090/S0002-9947-2011-05536-4
month: '03'
oa: 1
oa_version: Published Version
page: 1571-1631
publication: Transactions of the American Mathematical Society
publication_identifier:
  eissn:
  - 1088-6850
  issn:
  - 0002-9947
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Scattering for the cubic Klein-Gordon equation in two space dimensions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 364
year: '2012'
...