{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:RHEHDHKYOV5Q2E5TGIUH7GUMKW","short_pith_number":"pith:RHEHDHKY","canonical_record":{"source":{"id":"1604.06126","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-20T21:40:37Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"76fb64f928b81d1ee507f0af413818f344ae22deda683de4c9194d95840f18c1","abstract_canon_sha256":"fc13b99409c9b048d56825af98e25c5b6bad45535eb93d7a8ed4f7d7320ba5c5"},"schema_version":"1.0"},"canonical_sha256":"89c8719d58757b0d13b332287f9a8c55a2aeda600da1950e99a154e5442fe8aa","source":{"kind":"arxiv","id":"1604.06126","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.06126","created_at":"2026-05-18T01:16:32Z"},{"alias_kind":"arxiv_version","alias_value":"1604.06126v1","created_at":"2026-05-18T01:16:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06126","created_at":"2026-05-18T01:16:32Z"},{"alias_kind":"pith_short_12","alias_value":"RHEHDHKYOV5Q","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RHEHDHKYOV5Q2E5T","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RHEHDHKY","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:RHEHDHKYOV5Q2E5TGIUH7GUMKW","target":"record","payload":{"canonical_record":{"source":{"id":"1604.06126","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-20T21:40:37Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"76fb64f928b81d1ee507f0af413818f344ae22deda683de4c9194d95840f18c1","abstract_canon_sha256":"fc13b99409c9b048d56825af98e25c5b6bad45535eb93d7a8ed4f7d7320ba5c5"},"schema_version":"1.0"},"canonical_sha256":"89c8719d58757b0d13b332287f9a8c55a2aeda600da1950e99a154e5442fe8aa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:32.558725Z","signature_b64":"0YjT7k6hjSw7pJAejad2LpZcGgjoKldd7iRma/k5tMHfO1RklX401RkijuE49+JEPpx2EuZZrAR4EVI0cCFHBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89c8719d58757b0d13b332287f9a8c55a2aeda600da1950e99a154e5442fe8aa","last_reissued_at":"2026-05-18T01:16:32.557945Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:32.557945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.06126","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":"2026-05-18T01:16:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"idaBuTPZ25aWVqXPJQO3BGdcXZpKoGQcPlm/qBJLJY4Z78cmHJQvbh0e9W3EJXp54f8SSs9z8ZKWpjpJH/zkDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T11:43:46.695058Z"},"content_sha256":"ab7044888b8a642b825f666a89eb036ac6571834ba2b43b552fb6151d89292ff","schema_version":"1.0","event_id":"sha256:ab7044888b8a642b825f666a89eb036ac6571834ba2b43b552fb6151d89292ff"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:RHEHDHKYOV5Q2E5TGIUH7GUMKW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Gabriele Grillo, Juan Luis V\\'azquez, Matteo Muratori","submitted_at":"2016-04-20T21:40:37Z","abstract_excerpt":"We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06126","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":"2026-05-18T01:16:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cBCAbP3wRPbLfOuqfcfmh9Cr4DwIlk1i5auDPdkrUBBbFJtpPyWEEQ/DmfZvN0wANYmlcXnwFqe1aDnwpjbNCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T11:43:46.695728Z"},"content_sha256":"f137d42560490391b1a29dc1f716768e844eb7e18c6708c98cb0dbb300bdb58e","schema_version":"1.0","event_id":"sha256:f137d42560490391b1a29dc1f716768e844eb7e18c6708c98cb0dbb300bdb58e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RHEHDHKYOV5Q2E5TGIUH7GUMKW/bundle.json","state_url":"https://pith.science/pith/RHEHDHKYOV5Q2E5TGIUH7GUMKW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RHEHDHKYOV5Q2E5TGIUH7GUMKW/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":"2026-05-30T11:43:46Z","links":{"resolver":"https://pith.science/pith/RHEHDHKYOV5Q2E5TGIUH7GUMKW","bundle":"https://pith.science/pith/RHEHDHKYOV5Q2E5TGIUH7GUMKW/bundle.json","state":"https://pith.science/pith/RHEHDHKYOV5Q2E5TGIUH7GUMKW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RHEHDHKYOV5Q2E5TGIUH7GUMKW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:RHEHDHKYOV5Q2E5TGIUH7GUMKW","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":"fc13b99409c9b048d56825af98e25c5b6bad45535eb93d7a8ed4f7d7320ba5c5","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-20T21:40:37Z","title_canon_sha256":"76fb64f928b81d1ee507f0af413818f344ae22deda683de4c9194d95840f18c1"},"schema_version":"1.0","source":{"id":"1604.06126","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.06126","created_at":"2026-05-18T01:16:32Z"},{"alias_kind":"arxiv_version","alias_value":"1604.06126v1","created_at":"2026-05-18T01:16:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06126","created_at":"2026-05-18T01:16:32Z"},{"alias_kind":"pith_short_12","alias_value":"RHEHDHKYOV5Q","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RHEHDHKYOV5Q2E5T","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RHEHDHKY","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:f137d42560490391b1a29dc1f716768e844eb7e18c6708c98cb0dbb300bdb58e","target":"graph","created_at":"2026-05-18T01:16:32Z","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 consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such ma","authors_text":"Gabriele Grillo, Juan Luis V\\'azquez, Matteo Muratori","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-20T21:40:37Z","title":"The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06126","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:ab7044888b8a642b825f666a89eb036ac6571834ba2b43b552fb6151d89292ff","target":"record","created_at":"2026-05-18T01:16:32Z","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":"fc13b99409c9b048d56825af98e25c5b6bad45535eb93d7a8ed4f7d7320ba5c5","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-20T21:40:37Z","title_canon_sha256":"76fb64f928b81d1ee507f0af413818f344ae22deda683de4c9194d95840f18c1"},"schema_version":"1.0","source":{"id":"1604.06126","kind":"arxiv","version":1}},"canonical_sha256":"89c8719d58757b0d13b332287f9a8c55a2aeda600da1950e99a154e5442fe8aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"89c8719d58757b0d13b332287f9a8c55a2aeda600da1950e99a154e5442fe8aa","first_computed_at":"2026-05-18T01:16:32.557945Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:32.557945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0YjT7k6hjSw7pJAejad2LpZcGgjoKldd7iRma/k5tMHfO1RklX401RkijuE49+JEPpx2EuZZrAR4EVI0cCFHBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:32.558725Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.06126","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ab7044888b8a642b825f666a89eb036ac6571834ba2b43b552fb6151d89292ff","sha256:f137d42560490391b1a29dc1f716768e844eb7e18c6708c98cb0dbb300bdb58e"],"state_sha256":"395735575a4f9edf9853b0dd43df477102530b019f46df0c8c69f20e74e6f10c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"djbICA7XffIZCkkBA6xq8e+8CAjnVcxN/1osdSTxwhdJICbqsbtZSYf342Dc+fa2XDvLZ6LzoGlrRbHC1g73Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T11:43:46.699263Z","bundle_sha256":"0b986e679c06279e4d2db79ee70a4c6ebf65b0f0770c7ce2479d3d1596975c4f"}}