{"paper":{"title":"Existence of finite time blow-up in Keller-Segel system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Federico Buseghin, Juan Davila, Manuel del Pino, Monica Musso","submitted_at":"2023-12-03T18:21:07Z","abstract_excerpt":"Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \n  $\\begin{equation} \\begin{cases} u_{t} =\\Delta u - \\nabla \\cdot(u \\nabla v) \\ \\ \\ \\text{in } \\mathbb{R}^2\\times(0,T),\\\\[5pt] v = (-\\Delta_{\\mathbb{R}^2})^{-1} u := \\displaystyle\\frac {1}{2\\pi} \\displaystyle\\int_{\\mathbb{R}^2} \\log \\frac {1}{|x-z|}u(z,t) dz, \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\star)\\\\[5pt] u(\\cdot ,0) = u_{0}^{\\star} \\ge 0 \\ \\ \\ \\text{in } \\mathbb{R}^2. \\end{cases} \\end{equation}$\n  We show that there exists $\\varepsilon>0$ such that for any $m$ satisfying $8\\pi<m\\le 8\\pi+\\varepsilon$ \n  and any $k$ given po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.01475","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.01475/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"}