{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:JHZNJGVHBMHB46YA3HVGZLAKZ6","short_pith_number":"pith:JHZNJGVH","canonical_record":{"source":{"id":"1104.1513","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-08T08:03:16Z","cross_cats_sorted":[],"title_canon_sha256":"be0b9d56e5fc4acf7420e38d5ab0fa2e0d0ac1ca413cbec76cb58c93d06ece53","abstract_canon_sha256":"1dde2ecd5d412f7193522ed2d6fc7291e37391df6e17b575ab7391f5b5c6ac2c"},"schema_version":"1.0"},"canonical_sha256":"49f2d49aa70b0e1e7b00d9ea6cac0acfa891d5ed6a372b21a664c411e3156a38","source":{"kind":"arxiv","id":"1104.1513","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.1513","created_at":"2026-05-18T04:01:23Z"},{"alias_kind":"arxiv_version","alias_value":"1104.1513v1","created_at":"2026-05-18T04:01:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.1513","created_at":"2026-05-18T04:01:23Z"},{"alias_kind":"pith_short_12","alias_value":"JHZNJGVHBMHB","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JHZNJGVHBMHB46YA","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JHZNJGVH","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:JHZNJGVHBMHB46YA3HVGZLAKZ6","target":"record","payload":{"canonical_record":{"source":{"id":"1104.1513","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-08T08:03:16Z","cross_cats_sorted":[],"title_canon_sha256":"be0b9d56e5fc4acf7420e38d5ab0fa2e0d0ac1ca413cbec76cb58c93d06ece53","abstract_canon_sha256":"1dde2ecd5d412f7193522ed2d6fc7291e37391df6e17b575ab7391f5b5c6ac2c"},"schema_version":"1.0"},"canonical_sha256":"49f2d49aa70b0e1e7b00d9ea6cac0acfa891d5ed6a372b21a664c411e3156a38","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:23.285401Z","signature_b64":"HOk7XQZ27TUADvMNJAyaV52EilX8pJxbL9CPfzKNVO+AQEsbwXsnvMTlLCFhfoN7yUmuH0qHjMC74IKARnT7CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49f2d49aa70b0e1e7b00d9ea6cac0acfa891d5ed6a372b21a664c411e3156a38","last_reissued_at":"2026-05-18T04:01:23.284913Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:23.284913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.1513","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":"2026-05-18T04:01:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f5kTJRPCVf7JIxAkFZGfb7FfERSO7nJBkjAEx7GvRlyR+6CapmY3MG2EA9hJXYMzI5bVWKDCJ7dd4HU9+DjIDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:42:19.822146Z"},"content_sha256":"be0a9c0015c94e154ab604750d1b427b1fce372cb51b6677b74b69ab17a51eb6","schema_version":"1.0","event_id":"sha256:be0a9c0015c94e154ab604750d1b427b1fce372cb51b6677b74b69ab17a51eb6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:JHZNJGVHBMHB46YA3HVGZLAKZ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positivity, decay, and extinction for a singular diffusion equation with gradient absorption","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"IMAR), Philippe Laurencot (IMT), Razvan Gabriel Iagar (IMT","submitted_at":"2011-04-08T08:03:16Z","abstract_excerpt":"We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \\partial_t u -\\Delta_{p}u+|\\nabla u|^{q}=0\\quad in\\;\\; (0,\\infty)\\times\\RR^N, where $N\\ge 1$, $p\\in(1,2)$, and $q>0$. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as $t\\to\\infty$ for $q>p-N/(N+1)$, optimal decay estimates as $t\\to\\infty$ for $p/2\\le q\\le p-N/(N+1)$, or extinction in finite time for $0 < q < p/2$. In addition, we show how the diffusion prevents extinction "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1513","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:01:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m9TOjz+o0DZ6LjxLkEKklLPQXwpU8Po0X8SGTOJqblGhDZ+lOlAWW0NPsHGEmdVv71fXEq1VE5uG7KRo0mklAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:42:19.822525Z"},"content_sha256":"593c1ee14410f7011f1555d2da0659b09858a334ee0fed376b60e06cd3b1a930","schema_version":"1.0","event_id":"sha256:593c1ee14410f7011f1555d2da0659b09858a334ee0fed376b60e06cd3b1a930"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JHZNJGVHBMHB46YA3HVGZLAKZ6/bundle.json","state_url":"https://pith.science/pith/JHZNJGVHBMHB46YA3HVGZLAKZ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JHZNJGVHBMHB46YA3HVGZLAKZ6/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":"2026-06-01T14:42:19Z","links":{"resolver":"https://pith.science/pith/JHZNJGVHBMHB46YA3HVGZLAKZ6","bundle":"https://pith.science/pith/JHZNJGVHBMHB46YA3HVGZLAKZ6/bundle.json","state":"https://pith.science/pith/JHZNJGVHBMHB46YA3HVGZLAKZ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JHZNJGVHBMHB46YA3HVGZLAKZ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JHZNJGVHBMHB46YA3HVGZLAKZ6","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":"1dde2ecd5d412f7193522ed2d6fc7291e37391df6e17b575ab7391f5b5c6ac2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-08T08:03:16Z","title_canon_sha256":"be0b9d56e5fc4acf7420e38d5ab0fa2e0d0ac1ca413cbec76cb58c93d06ece53"},"schema_version":"1.0","source":{"id":"1104.1513","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.1513","created_at":"2026-05-18T04:01:23Z"},{"alias_kind":"arxiv_version","alias_value":"1104.1513v1","created_at":"2026-05-18T04:01:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.1513","created_at":"2026-05-18T04:01:23Z"},{"alias_kind":"pith_short_12","alias_value":"JHZNJGVHBMHB","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JHZNJGVHBMHB46YA","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JHZNJGVH","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:593c1ee14410f7011f1555d2da0659b09858a334ee0fed376b60e06cd3b1a930","target":"graph","created_at":"2026-05-18T04:01:23Z","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"},"paper":{"abstract_excerpt":"We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \\partial_t u -\\Delta_{p}u+|\\nabla u|^{q}=0\\quad in\\;\\; (0,\\infty)\\times\\RR^N, where $N\\ge 1$, $p\\in(1,2)$, and $q>0$. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as $t\\to\\infty$ for $q>p-N/(N+1)$, optimal decay estimates as $t\\to\\infty$ for $p/2\\le q\\le p-N/(N+1)$, or extinction in finite time for $0 < q < p/2$. In addition, we show how the diffusion prevents extinction ","authors_text":"IMAR), Philippe Laurencot (IMT), Razvan Gabriel Iagar (IMT","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-08T08:03:16Z","title":"Positivity, decay, and extinction for a singular diffusion equation with gradient absorption"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1513","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:be0a9c0015c94e154ab604750d1b427b1fce372cb51b6677b74b69ab17a51eb6","target":"record","created_at":"2026-05-18T04:01:23Z","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":"1dde2ecd5d412f7193522ed2d6fc7291e37391df6e17b575ab7391f5b5c6ac2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-08T08:03:16Z","title_canon_sha256":"be0b9d56e5fc4acf7420e38d5ab0fa2e0d0ac1ca413cbec76cb58c93d06ece53"},"schema_version":"1.0","source":{"id":"1104.1513","kind":"arxiv","version":1}},"canonical_sha256":"49f2d49aa70b0e1e7b00d9ea6cac0acfa891d5ed6a372b21a664c411e3156a38","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49f2d49aa70b0e1e7b00d9ea6cac0acfa891d5ed6a372b21a664c411e3156a38","first_computed_at":"2026-05-18T04:01:23.284913Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:01:23.284913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HOk7XQZ27TUADvMNJAyaV52EilX8pJxbL9CPfzKNVO+AQEsbwXsnvMTlLCFhfoN7yUmuH0qHjMC74IKARnT7CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:01:23.285401Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.1513","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be0a9c0015c94e154ab604750d1b427b1fce372cb51b6677b74b69ab17a51eb6","sha256:593c1ee14410f7011f1555d2da0659b09858a334ee0fed376b60e06cd3b1a930"],"state_sha256":"b02d47aa2f66644d9ee79d87be91ba97a443484e87c246598aa113657d001bfc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8u4ZYtQXc+1YfwBzDXwHTZkNmwtHrRoXcZM5ZtmnGYl0sbpjKi+3zJz76FGsGvmflUsUJwF21vkaxea46YQhAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T14:42:19.824615Z","bundle_sha256":"e7e73892487031fa4e87bfd142d7d5a828fc4e470e1eab0b5a19307d1107f88a"}}