{"paper":{"title":"The dynamics of the 3D radial NLS with the combined terms","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Lifeng Zhao","submitted_at":"2011-11-29T02:37:43Z","abstract_excerpt":"In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\\\"{o}dinger equation (NLS) with the combined terms\niu_t + \\Delta u = -|u|^4u + |u|^2u \\tag{CNLS}\nin the energy space $H^1(\\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \\Delta u = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \\cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \\cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6671","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}