{"paper":{"title":"Scattering theory for the defocusing fourth-order Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Jiqiang Zheng","submitted_at":"2012-11-20T05:33:33Z","abstract_excerpt":"In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schr\\\"odinger equation (FNLS) $iu_t+\\Delta^2 u+|u|^pu=0$ in dimension $d\\geq9$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\\in L_t^\\infty(I;\\dot H^{s_c}_x(\\R^d))$ with all $s_c:=\\frac{d}2-\\frac4p\\geq1$ if $p$ is an even integer or $s_c\\in[1,2+p)$ otherwise, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical and energy-subcritical nonlinear Schr\\\""},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4668","kind":"arxiv","version":2},"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"}