{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:GM3RITTPAWDERJSSOWJT3WDBK2","short_pith_number":"pith:GM3RITTP","canonical_record":{"source":{"id":"0801.2310","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2008-01-15T14:34:24Z","cross_cats_sorted":[],"title_canon_sha256":"afe2aa26d99eedac9c1dfd7c7b42116ec13a7c8a673f6729dae3a2cec07b1fe8","abstract_canon_sha256":"bf2a1a925e666c1e973b7aaad54abd5c4b4a51bde8414d16b76d18253673d1e5"},"schema_version":"1.0"},"canonical_sha256":"3337144e6f058648a65275933dd86156bc8bfaa228c0439bcd222d380392f475","source":{"kind":"arxiv","id":"0801.2310","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0801.2310","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"arxiv_version","alias_value":"0801.2310v1","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0801.2310","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"pith_short_12","alias_value":"GM3RITTPAWDE","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"pith_short_16","alias_value":"GM3RITTPAWDERJSS","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"pith_short_8","alias_value":"GM3RITTP","created_at":"2026-07-04T15:06:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:GM3RITTPAWDERJSSOWJT3WDBK2","target":"record","payload":{"canonical_record":{"source":{"id":"0801.2310","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2008-01-15T14:34:24Z","cross_cats_sorted":[],"title_canon_sha256":"afe2aa26d99eedac9c1dfd7c7b42116ec13a7c8a673f6729dae3a2cec07b1fe8","abstract_canon_sha256":"bf2a1a925e666c1e973b7aaad54abd5c4b4a51bde8414d16b76d18253673d1e5"},"schema_version":"1.0"},"canonical_sha256":"3337144e6f058648a65275933dd86156bc8bfaa228c0439bcd222d380392f475","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:06:51.340553Z","signature_b64":"ZN8Ihj9Gtkj4etujAMOpX+dNd984Q/qBkrZuCieigeSAfOCiAFa0akoSJYs7LqR+54zs50ZIE3fJV8YyJhgKBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3337144e6f058648a65275933dd86156bc8bfaa228c0439bcd222d380392f475","last_reissued_at":"2026-07-04T15:06:51.340162Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:06:51.340162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0801.2310","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:06:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8MIN+tvZLhC2okqlHCHYwr41y3kdE9SzeOo+ZXAoTK21Xlr2LN9VUO7uLyTRJqxliu9nNImFKozw9bth5DvzCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-01T10:34:36.258544Z"},"content_sha256":"5d9bd6128450a04ce60f5e7b8b945e902b0ad008efd9995d48f56703b71e6568","schema_version":"1.0","event_id":"sha256:5d9bd6128450a04ce60f5e7b8b945e902b0ad008efd9995d48f56703b71e6568"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:GM3RITTPAWDERJSSOWJT3WDBK2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adrien Blanchet, Jos\\'e Antonio Carrillo, Philippe Lauren\\c{c}ot","submitted_at":"2008-01-15T14:34:24Z","abstract_excerpt":"This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \\in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0801.2310","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0801.2310/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:06:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W2o33qS7hXXTYZqIwZmTZB8onGDqNl4HwwftKfHriIgMvYbokSeUj3MGBInDVhoXq1pMpEt+ajey/OccyNJRDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-01T10:34:36.259057Z"},"content_sha256":"c40f8bc50b6cf4fa54d9bf2cbf46d007ce5b2837ba5a505d7492c6f23047d0a9","schema_version":"1.0","event_id":"sha256:c40f8bc50b6cf4fa54d9bf2cbf46d007ce5b2837ba5a505d7492c6f23047d0a9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GM3RITTPAWDERJSSOWJT3WDBK2/bundle.json","state_url":"https://pith.science/pith/GM3RITTPAWDERJSSOWJT3WDBK2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GM3RITTPAWDERJSSOWJT3WDBK2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-01T10:34:36Z","links":{"resolver":"https://pith.science/pith/GM3RITTPAWDERJSSOWJT3WDBK2","bundle":"https://pith.science/pith/GM3RITTPAWDERJSSOWJT3WDBK2/bundle.json","state":"https://pith.science/pith/GM3RITTPAWDERJSSOWJT3WDBK2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GM3RITTPAWDERJSSOWJT3WDBK2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:GM3RITTPAWDERJSSOWJT3WDBK2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"bf2a1a925e666c1e973b7aaad54abd5c4b4a51bde8414d16b76d18253673d1e5","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2008-01-15T14:34:24Z","title_canon_sha256":"afe2aa26d99eedac9c1dfd7c7b42116ec13a7c8a673f6729dae3a2cec07b1fe8"},"schema_version":"1.0","source":{"id":"0801.2310","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0801.2310","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"arxiv_version","alias_value":"0801.2310v1","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0801.2310","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"pith_short_12","alias_value":"GM3RITTPAWDE","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"pith_short_16","alias_value":"GM3RITTPAWDERJSS","created_at":"2026-07-04T15:06:51Z"},{"alias_kind":"pith_short_8","alias_value":"GM3RITTP","created_at":"2026-07-04T15:06:51Z"}],"graph_snapshots":[{"event_id":"sha256:c40f8bc50b6cf4fa54d9bf2cbf46d007ce5b2837ba5a505d7492c6f23047d0a9","target":"graph","created_at":"2026-07-04T15:06:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0801.2310/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \\in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the ex","authors_text":"Adrien Blanchet, Jos\\'e Antonio Carrillo, Philippe Lauren\\c{c}ot","cross_cats":[],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2008-01-15T14:34:24Z","title":"Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0801.2310","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5d9bd6128450a04ce60f5e7b8b945e902b0ad008efd9995d48f56703b71e6568","target":"record","created_at":"2026-07-04T15:06:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"bf2a1a925e666c1e973b7aaad54abd5c4b4a51bde8414d16b76d18253673d1e5","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2008-01-15T14:34:24Z","title_canon_sha256":"afe2aa26d99eedac9c1dfd7c7b42116ec13a7c8a673f6729dae3a2cec07b1fe8"},"schema_version":"1.0","source":{"id":"0801.2310","kind":"arxiv","version":1}},"canonical_sha256":"3337144e6f058648a65275933dd86156bc8bfaa228c0439bcd222d380392f475","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3337144e6f058648a65275933dd86156bc8bfaa228c0439bcd222d380392f475","first_computed_at":"2026-07-04T15:06:51.340162Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:06:51.340162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZN8Ihj9Gtkj4etujAMOpX+dNd984Q/qBkrZuCieigeSAfOCiAFa0akoSJYs7LqR+54zs50ZIE3fJV8YyJhgKBw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:06:51.340553Z","signed_message":"canonical_sha256_bytes"},"source_id":"0801.2310","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d9bd6128450a04ce60f5e7b8b945e902b0ad008efd9995d48f56703b71e6568","sha256:c40f8bc50b6cf4fa54d9bf2cbf46d007ce5b2837ba5a505d7492c6f23047d0a9"],"state_sha256":"cf0da6ac5f3aba4559013a1b01b7dd4b48fedb40c02d14adf4b490381ff76d59"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P3bdrVgvSW1KcqkL8ERQEbbEGW6u7NTFhaIF9L+L9PmEbPM7IMLTzASs8oeezwGJLkYnLWORFJtduEs2YDbcDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-01T10:34:36.265436Z","bundle_sha256":"2b9d6db5b5dfd7278dfdfcbbd4bfeedf86a1c3c7b4a55b00d54f1574d37acbb8"}}