{"paper":{"title":"Scattering and Blow up for the Two Dimensional Focusing Quintic Nonlinear Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cristi Guevara, Fernando Carreon","submitted_at":"2012-03-27T21:16:13Z","abstract_excerpt":"Using the concentration-compactness method and the localized virial type arguments, we study the behavior of $H^1$ solutions to the focusing quintic NLS in $\\R^2$, namely, $$i \\partial_t u+\\Delta u+|u|^4u=0,\\quad\\quad (x, t) \\in \\R^2\\times\\R.$$\n  Denoting by $M[u]$ and $E[u]$, the mass and energy of a solution $u,$ respectively, and $Q$ the ground state solution to $-Q+\\Delta Q+ |Q|^4Q=0$, and assuming $M[u]E[u] <M[Q]E[Q]$, we characterize the threshold for global versus finite time existence. Moreover, we show scattering for global existing time solutions and finite or \"weak\" blow up for the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6089","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"}