{"paper":{"title":"Existence and stability of infinite time blow-up in the Keller-Segel system","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Dolbeault, Juan Davila, Juncheng Wei, Manuel del Pino, Monica Musso","submitted_at":"2019-11-27T20:46:45Z","abstract_excerpt":"Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \\begin{equation}\\tag{$\\ast$} \\label{ks0} \\left\\{ \\begin{aligned} u_t =&\\; \\Delta u - \\nabla \\cdot(u \\nabla v) \\quad in {\\mathbb R}^2\\times(0,\\infty),\\\\ v =&\\; (-\\Delta_{\\R^2})^{-1} u := \\frac 1{2\\pi} \\int_{R^2} \\, \\log \\frac 1{|x-z|}\\,u(z,t)\\, dz, \\\\ & \\qquad\\ u(\\cdot ,0) = u_0 \\geq 0\\quad\\hbox{in } R^2. \\end{aligned} \\right. \\end{equation} We consider the {\\em critical mass case} $\\int_{R^2} u_0(x)\\, dx = 8\\pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion tow"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.12417","kind":"arxiv","version":3},"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/1911.12417/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"}