{"paper":{"title":"New functional inequalities with applications to the arctan-fast diffusion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alejandro Ortega, Martina Magliocca, Rafael Granero-Belinch\\'on","submitted_at":"2024-03-15T16:49:31Z","abstract_excerpt":"In this paper, we prove a couple of new nonlinear functional inequalities of Sobolev type akin to the logarithmic Sobolev inequality. In particular, one of the inequalities reads $$ \\int_{\\mathbb{S}^1}\\arctan\\left(\\frac{\\partial_x u}{u}\\right)\\partial_xu \\,dx\\geq \\arctan\\left(\\|u(t)\\|_{\\dot{W}^{1,1}(\\mathbb{S}^1)}\\right)\\|u(t)\\|_{\\dot{W}^{1,1}(\\mathbb{S}^1)}. $$ Then, these inequalities are used in the study of the nonlinear \\emph{arctan}-fast diffusion equation $$ \\partial_t u-\\partial_x\\arctan\\left(\\frac{\\partial_x u}{u}\\right)=0. $$ For this highly nonlinear PDE we establish a number of wel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.10458","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.10458/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}