{"paper":{"title":"The supercritical generalized KdV equation: Global well-posedness in the energy space and below","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ademir Pastor, Felipe Linares, Luiz Gustavo Farah","submitted_at":"2010-09-16T18:10:09Z","abstract_excerpt":"We consider the generalized Korteweg-de Vries (gKdV) equation\n  $\\partial_t u+\\partial_x^3u+\\mu\\partial_x(u^{k+1})=0$, where $k\\geq5$ is an integer number and $\\mu=\\pm1$. In the focusing case ($\\mu=1$), we show that if the initial data $u_0$ belongs to $H^1(\\R)$ and satisfies $E(u_0)^{s_k} M(u_0)^{1-s_k} < E(Q)^{s_k} M(Q)^{1-s_k}$, $E(u_0)\\geq0$, and $\\|\\partial_x u_0\\|_{L^2}^{s_k}\\|u_0\\|_{L^2}^{1-s_k} < \\|\\partial_x Q\\|_{L^2}^{s_k}\\|Q\\|_{L^2}^{1-s_k}$, where $M(u)$ and $E(u)$ are the mass and energy, then the corresponding solution is global in $H^1(\\R)$. Here, $s_k=\\frac{(k-4)}{2k}$ and $Q$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3234","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"}