{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:ASRXBVZTJAQEOEQFCW7ASPIQOR","short_pith_number":"pith:ASRXBVZT","schema_version":"1.0","canonical_sha256":"04a370d733482047120515be093d107474029ded5151a4478dfd65ea188e1cf9","source":{"kind":"arxiv","id":"1106.5345","version":1},"attestation_state":"computed","paper":{"title":"Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Winkler, Youshan Tao","submitted_at":"2011-06-27T10:11:57Z","abstract_excerpt":"We consider the quasilinear parabolic-parabolic Keller-Segel system $$\n u_t=\\nabla \\cdot (D(u)\\nabla u) - \\nabla \\cdot (S(u)\\nabla v),\n  \\qquad x\\in\\Omega, \\ t>0,\n v_t=\\Delta v -v + u,\n  x\\in\\Omega, \\ t>0,\n$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\\Omega\\subset\\R^n$ with $n\\ge 2$.\nIt is proved that if $\\frac{S(u)}{D(u)}\\le cu^{\\alpha}$ with $\\alpha<\\frac{2}{n}$ and some constant $c>0$ for all $u>1$ and some further technical conditions are fulfilled, then the classical solutions to the above system are uniformly-in-time bounded. This boundedness result is opt"},"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":"1106.5345","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-06-27T10:11:57Z","cross_cats_sorted":[],"title_canon_sha256":"75ac18d49055ee71a2ddb85c9d3da9d90eb535dea56f03081b276533d3b0475e","abstract_canon_sha256":"0c8d1865aa68df4327c0cb6290069ee62e02f037fe608a0729d52e898c426ba7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:13.771343Z","signature_b64":"z4kBXgi2iDUv4aQhQTkBN6uQHykQUWasfOlcvBdKzh/9VfMJqjE6M6NUUzpWj2vcnTEzags2bCJ1dol6FcpyDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04a370d733482047120515be093d107474029ded5151a4478dfd65ea188e1cf9","last_reissued_at":"2026-05-18T04:19:13.770918Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:13.770918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Winkler, Youshan Tao","submitted_at":"2011-06-27T10:11:57Z","abstract_excerpt":"We consider the quasilinear parabolic-parabolic Keller-Segel system $$\n u_t=\\nabla \\cdot (D(u)\\nabla u) - \\nabla \\cdot (S(u)\\nabla v),\n  \\qquad x\\in\\Omega, \\ t>0,\n v_t=\\Delta v -v + u,\n  x\\in\\Omega, \\ t>0,\n$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\\Omega\\subset\\R^n$ with $n\\ge 2$.\nIt is proved that if $\\frac{S(u)}{D(u)}\\le cu^{\\alpha}$ with $\\alpha<\\frac{2}{n}$ and some constant $c>0$ for all $u>1$ and some further technical conditions are fulfilled, then the classical solutions to the above system are uniformly-in-time bounded. This boundedness result is opt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5345","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":"1106.5345","created_at":"2026-05-18T04:19:13.770977+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.5345v1","created_at":"2026-05-18T04:19:13.770977+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5345","created_at":"2026-05-18T04:19:13.770977+00:00"},{"alias_kind":"pith_short_12","alias_value":"ASRXBVZTJAQE","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"ASRXBVZTJAQEOEQF","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"ASRXBVZT","created_at":"2026-05-18T12:26:24.575870+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR","json":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR.json","graph_json":"https://pith.science/api/pith-number/ASRXBVZTJAQEOEQFCW7ASPIQOR/graph.json","events_json":"https://pith.science/api/pith-number/ASRXBVZTJAQEOEQFCW7ASPIQOR/events.json","paper":"https://pith.science/paper/ASRXBVZT"},"agent_actions":{"view_html":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR","download_json":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR.json","view_paper":"https://pith.science/paper/ASRXBVZT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.5345&json=true","fetch_graph":"https://pith.science/api/pith-number/ASRXBVZTJAQEOEQFCW7ASPIQOR/graph.json","fetch_events":"https://pith.science/api/pith-number/ASRXBVZTJAQEOEQFCW7ASPIQOR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR/action/storage_attestation","attest_author":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR/action/author_attestation","sign_citation":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR/action/citation_signature","submit_replication":"https://pith.science/pith/ASRXBVZTJAQEOEQFCW7ASPIQOR/action/replication_record"}},"created_at":"2026-05-18T04:19:13.770977+00:00","updated_at":"2026-05-18T04:19:13.770977+00:00"}