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\\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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1112.4156","created_at":"2026-05-18T04:06:06.929654+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.4156v1","created_at":"2026-05-18T04:06:06.929654+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4156","created_at":"2026-05-18T04:06:06.929654+00:00"},{"alias_kind":"pith_short_12","alias_value":"52FIHRPNXS2U","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"52FIHRPNXS2U7WIU","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"52FIHRPN","created_at":"2026-05-18T12:26:20.644004+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.13300","citing_title":"Refined temporal asymptotics near blow-up points in the planar Keller-Segel system","ref_index":38,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY","json":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY.json","graph_json":"https://pith.science/api/pith-number/52FIHRPNXS2U7WIUD2WWHMDIYY/graph.json","events_json":"https://pith.science/api/pith-number/52FIHRPNXS2U7WIUD2WWHMDIYY/events.json","paper":"https://pith.science/paper/52FIHRPN"},"agent_actions":{"view_html":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY","download_json":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY.json","view_paper":"https://pith.science/paper/52FIHRPN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.4156&json=true","fetch_graph":"https://pith.science/api/pith-number/52FIHRPNXS2U7WIUD2WWHMDIYY/graph.json","fetch_events":"https://pith.science/api/pith-number/52FIHRPNXS2U7WIUD2WWHMDIYY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY/action/storage_attestation","attest_author":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY/action/author_attestation","sign_citation":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY/action/citation_signature","submit_replication":"https://pith.science/pith/52FIHRPNXS2U7WIUD2WWHMDIYY/action/replication_record"}},"created_at":"2026-05-18T04:06:06.929654+00:00","updated_at":"2026-05-18T04:06:06.929654+00:00"}