{"paper":{"title":"Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"H. Zaag, N. Nouaili, V. T. Nguyen","submitted_at":"2023-09-25T08:02:57Z","abstract_excerpt":"We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system\n  $\\partial_t u = \\Delta u - \\nabla \\cdot (u \\nabla \\mathcal{K}_u), \\quad -\\Delta \\mathcal{K}_u = u \\quad \\textup{in}\\;\\; \\mathbb{R}^d,\\; d = 3,4,$\n  and derive the final blowup profile\n  $ u(r,T) \\sim c_d \\frac{|\\log r|^\\frac{d-2}{d}}{r^2} \\quad \\textup{as}\\;\\; r \\to 0, \\;\\; c_d > 0.$\n  To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.13932","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/2309.13932/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"}