{"date_published":"2012-03-01T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.1090/S0002-9947-2011-05536-4","open_access":"1"}],"_id":"22022","date_updated":"2026-06-22T09:19:43Z","language":[{"iso":"eng"}],"quality_controlled":"1","year":"2012","month":"03","status":"public","page":"1571-1631","arxiv":1,"title":"Scattering for the cubic Klein-Gordon equation in two space dimensions","volume":364,"extern":"1","doi":"10.1090/s0002-9947-2011-05536-4","publication_status":"published","publication":"Transactions of the American Mathematical Society","abstract":[{"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.","lang":"eng"}],"oa_version":"Published Version","publication_identifier":{"eissn":["1088-6850"],"issn":["0002-9947"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"American Mathematical Society","article_type":"original","date_created":"2026-06-19T07:31:10Z","OA_place":"publisher","day":"01","intvolume":"       364","das_tickbox":"1","oa":1,"article_processing_charge":"No","type":"journal_article","citation":{"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.","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>","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>.","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>.","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>","short":"R. Killip, B. Stovall, M. Vişan, Transactions of the American Mathematical Society 364 (2012) 1571–1631."},"issue":"3","external_id":{"arxiv":["1008.2712"]},"scopus_import":"1","author":[{"first_name":"Rowan","last_name":"Killip","full_name":"Killip, Rowan"},{"full_name":"Stovall, Betsy","last_name":"Stovall","first_name":"Betsy"},{"last_name":"Visan","full_name":"Visan, Monica","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"}],"OA_type":"free access"}