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(-\\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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1911.12417","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2019-11-27T20:46:45Z","cross_cats_sorted":[],"title_canon_sha256":"ea814ebb06a98743babd99ccc250e978606de66a1c59d2000444b6ac76ce0dce","abstract_canon_sha256":"8b24253a5d591cd2f60d7b5a70d73a48414b3b36f545630eb2b08d8432fddfba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:41:48.928560Z","signature_b64":"9ogIK0o9Q4Wgf32LjOZbJjFyXP8UcnwkfKURYoPdk3W4iM6NFjGixrYoHV9HHBdCYU/NnnwEzjrhPLxpeb23DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"998bdbecef1d5f8419579fb0b7fcefb9de0dc611a806d9e59d0ab6746eaf99fb","last_reissued_at":"2026-07-05T05:41:48.928132Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:41:48.928132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1911.12417","created_at":"2026-07-05T05:41:48.928188+00:00"},{"alias_kind":"arxiv_version","alias_value":"1911.12417v3","created_at":"2026-07-05T05:41:48.928188+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.12417","created_at":"2026-07-05T05:41:48.928188+00:00"},{"alias_kind":"pith_short_12","alias_value":"TGF5X3HPDVPY","created_at":"2026-07-05T05:41:48.928188+00:00"},{"alias_kind":"pith_short_16","alias_value":"TGF5X3HPDVPYIGKX","created_at":"2026-07-05T05:41:48.928188+00:00"},{"alias_kind":"pith_short_8","alias_value":"TGF5X3HP","created_at":"2026-07-05T05:41:48.928188+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2508.08103","citing_title":"Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system","ref_index":28,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH","json":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH.json","graph_json":"https://pith.science/api/pith-number/TGF5X3HPDVPYIGKXT6YLP7HPXH/graph.json","events_json":"https://pith.science/api/pith-number/TGF5X3HPDVPYIGKXT6YLP7HPXH/events.json","paper":"https://pith.science/paper/TGF5X3HP"},"agent_actions":{"view_html":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH","download_json":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH.json","view_paper":"https://pith.science/paper/TGF5X3HP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1911.12417&json=true","fetch_graph":"https://pith.science/api/pith-number/TGF5X3HPDVPYIGKXT6YLP7HPXH/graph.json","fetch_events":"https://pith.science/api/pith-number/TGF5X3HPDVPYIGKXT6YLP7HPXH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH/action/storage_attestation","attest_author":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH/action/author_attestation","sign_citation":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH/action/citation_signature","submit_replication":"https://pith.science/pith/TGF5X3HPDVPYIGKXT6YLP7HPXH/action/replication_record"}},"created_at":"2026-07-05T05:41:48.928188+00:00","updated_at":"2026-07-05T05:41:48.928188+00:00"}