{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:X4MTJM3DLSSVFZK5F56SX3BLR5","short_pith_number":"pith:X4MTJM3D","schema_version":"1.0","canonical_sha256":"bf1934b3635ca552e55d2f7d2bec2b8f6f77b1b4d60fac881a3aedd248ccb693","source":{"kind":"arxiv","id":"1505.07790","version":3},"attestation_state":"computed","paper":{"title":"Gradient estimates of Hamilton - Souplet - Zhang type for a general heat equation on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nguyen Ngoc Khanh, Nguyen Thac Dung","submitted_at":"2015-05-28T18:33:59Z","abstract_excerpt":"The purpose of this paper is to study gradient estimate of Hamilton - Souplet - Zhang type for the general heat equation $$ u_t=\\Delta_V u + au\\log u+bu $$ on noncompact Riemannian manifolds. As its application, we show a Harnak inequality for the heat solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extention and improvement of the work of Souplet - Zhang (\\cite{SZ}), Ruan (\\cite{Ruan}), Yi Li (\\cite{Yili}) and Huang-Ma (\\cite{HM})."},"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":"1505.07790","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-05-28T18:33:59Z","cross_cats_sorted":[],"title_canon_sha256":"fe73299bb42bd6b35ce6d254b1140fa63bb4b9f0683f2acb58f09dc6a9185cff","abstract_canon_sha256":"443623ce78ae914cd1918c0fc0e80fdca960a5fcd90e699e8e42d2245215e8ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:08.002187Z","signature_b64":"73sE+29m7I54IkSfMxgYt4qsBT1BS9Lb/sFvx05TzOOzPFhtcHoIjMQW+YtS8MlUssIc3Crb1AN3aTxvF4taDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf1934b3635ca552e55d2f7d2bec2b8f6f77b1b4d60fac881a3aedd248ccb693","last_reissued_at":"2026-05-18T01:32:08.001451Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:08.001451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gradient estimates of Hamilton - Souplet - Zhang type for a general heat equation on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nguyen Ngoc Khanh, Nguyen Thac Dung","submitted_at":"2015-05-28T18:33:59Z","abstract_excerpt":"The purpose of this paper is to study gradient estimate of Hamilton - Souplet - Zhang type for the general heat equation $$ u_t=\\Delta_V u + au\\log u+bu $$ on noncompact Riemannian manifolds. As its application, we show a Harnak inequality for the heat solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extention and improvement of the work of Souplet - Zhang (\\cite{SZ}), Ruan (\\cite{Ruan}), Yi Li (\\cite{Yili}) and Huang-Ma (\\cite{HM})."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07790","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":""},"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":"1505.07790","created_at":"2026-05-18T01:32:08.001570+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.07790v3","created_at":"2026-05-18T01:32:08.001570+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.07790","created_at":"2026-05-18T01:32:08.001570+00:00"},{"alias_kind":"pith_short_12","alias_value":"X4MTJM3DLSSV","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"X4MTJM3DLSSVFZK5","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"X4MTJM3D","created_at":"2026-05-18T12:29:47.479230+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/X4MTJM3DLSSVFZK5F56SX3BLR5","json":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5.json","graph_json":"https://pith.science/api/pith-number/X4MTJM3DLSSVFZK5F56SX3BLR5/graph.json","events_json":"https://pith.science/api/pith-number/X4MTJM3DLSSVFZK5F56SX3BLR5/events.json","paper":"https://pith.science/paper/X4MTJM3D"},"agent_actions":{"view_html":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5","download_json":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5.json","view_paper":"https://pith.science/paper/X4MTJM3D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.07790&json=true","fetch_graph":"https://pith.science/api/pith-number/X4MTJM3DLSSVFZK5F56SX3BLR5/graph.json","fetch_events":"https://pith.science/api/pith-number/X4MTJM3DLSSVFZK5F56SX3BLR5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5/action/storage_attestation","attest_author":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5/action/author_attestation","sign_citation":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5/action/citation_signature","submit_replication":"https://pith.science/pith/X4MTJM3DLSSVFZK5F56SX3BLR5/action/replication_record"}},"created_at":"2026-05-18T01:32:08.001570+00:00","updated_at":"2026-05-18T01:32:08.001570+00:00"}