{"paper":{"title":"Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Winkler","submitted_at":"2011-12-18T13:42:57Z","abstract_excerpt":"We study the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system\nu_t=\\Delta u - \\nabla \\cdot (u\\nabla v), \\qquad x\\in\\Omega, \\ t>0,\n[1mm] v_t=\\Delta v-v+u, \\qquad x\\in\\Omega, \\ t>0,\nwhere $\\Omega$ is a ball in ${\\mathbb{R}}^n$ with $n\\ge 3$.\n  It is proved that for any prescribed $m>0$ there exist radially symmetric positive initial data\n$(u_0,v_0) \\in C^0(\\bar\\Omega) \\times W^{1,\\infty}(\\Omega)$ with $\\int_\\Omega u_0=m$\nsuch that the corresponding solution blows up in finite time.\n  Moreover, by providing an essentially explicit blow-up criterion it is shown tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4156","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"}