{"date_published":"2007-08-29T00:00:00Z","OA_type":"green","author":[{"full_name":"Tao, Terence","first_name":"Terence","last_name":"Tao"},{"last_name":"Visan","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"},{"first_name":"Xiaoyi","last_name":"Zhang","full_name":"Zhang, Xiaoyi"}],"extern":"1","intvolume":"        32","keyword":["Energy-critical","Mass-critical","Nonlinear Schrödinger equation","Wellposedness"],"status":"public","publication_status":"published","publisher":"Informa UK Limited","article_type":"original","date_updated":"2026-06-25T08:04:20Z","day":"29","year":"2007","abstract":[{"text":"We undertake a comprehensive study of the nonlinear Schrödinger equation (mathematical formular) where u(t, x) is a complex-valued function in spacetime R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical formular) . The results at the endpoint p1= 4/n are conditional on a conjectured global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint).\r\nAs an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H1/x for solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985).","lang":"eng"}],"publication":"Communications in Partial Differential Equations","volume":32,"mathsc":["35Q55"],"title":"The nonlinear Schrödinger equation with combined power-type nonlinearities","arxiv":1,"article_processing_charge":"No","external_id":{"arxiv":["math/0511070"]},"page":"1281-1343","doi":"10.1080/03605300701588805","date_created":"2026-06-19T07:49:46Z","_id":"22047","das_tickbox":"1","publication_identifier":{"eissn":["1532-4133"],"issn":["0360-5302"]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.math/0511070"}],"scopus_import":"1","citation":{"ama":"Tao T, Vişan M, Zhang X. The nonlinear Schrödinger equation with combined power-type nonlinearities. <i>Communications in Partial Differential Equations</i>. 2007;32(8):1281-1343. doi:<a href=\"https://doi.org/10.1080/03605300701588805\">10.1080/03605300701588805</a>","chicago":"Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial Differential Equations</i>. Informa UK Limited, 2007. <a href=\"https://doi.org/10.1080/03605300701588805\">https://doi.org/10.1080/03605300701588805</a>.","short":"T. Tao, M. Vişan, X. Zhang, Communications in Partial Differential Equations 32 (2007) 1281–1343.","ieee":"T. Tao, M. Vişan, and X. Zhang, “The nonlinear Schrödinger equation with combined power-type nonlinearities,” <i>Communications in Partial Differential Equations</i>, vol. 32, no. 8. Informa UK Limited, pp. 1281–1343, 2007.","mla":"Tao, Terence, et al. “The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial Differential Equations</i>, vol. 32, no. 8, Informa UK Limited, 2007, pp. 1281–343, doi:<a href=\"https://doi.org/10.1080/03605300701588805\">10.1080/03605300701588805</a>.","apa":"Tao, T., Vişan, M., &#38; Zhang, X. (2007). The nonlinear Schrödinger equation with combined power-type nonlinearities. <i>Communications in Partial Differential Equations</i>. Informa UK Limited. <a href=\"https://doi.org/10.1080/03605300701588805\">https://doi.org/10.1080/03605300701588805</a>","ista":"Tao T, Vişan M, Zhang X. 2007. The nonlinear Schrödinger equation with combined power-type nonlinearities. Communications in Partial Differential Equations. 32(8), 1281–1343."},"month":"08","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","issue":"8","language":[{"iso":"eng"}],"quality_controlled":"1","OA_place":"repository","oa":1,"oa_version":"Preprint"}