{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:3LQQCQPSNX26GVKND7ED45BMDT","short_pith_number":"pith:3LQQCQPS","canonical_record":{"source":{"id":"1906.08503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T08:53:20Z","cross_cats_sorted":[],"title_canon_sha256":"1b425810c1e31cbb63083cfe698c233bfb8d6fe3e9c94e013ef69c89239829b1","abstract_canon_sha256":"f9154acd0579293c52d4f793d3e34c95d8d6030ced3e4b90ada7cf28deae7e61"},"schema_version":"1.0"},"canonical_sha256":"dae10141f26df5e3554d1fc83e742c1cc422c2352a67803715944bbbece4adf5","source":{"kind":"arxiv","id":"1906.08503","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.08503","created_at":"2026-05-17T23:42:52Z"},{"alias_kind":"arxiv_version","alias_value":"1906.08503v1","created_at":"2026-05-17T23:42:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.08503","created_at":"2026-05-17T23:42:52Z"},{"alias_kind":"pith_short_12","alias_value":"3LQQCQPSNX26","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"3LQQCQPSNX26GVKN","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"3LQQCQPS","created_at":"2026-05-18T12:33:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:3LQQCQPSNX26GVKND7ED45BMDT","target":"record","payload":{"canonical_record":{"source":{"id":"1906.08503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T08:53:20Z","cross_cats_sorted":[],"title_canon_sha256":"1b425810c1e31cbb63083cfe698c233bfb8d6fe3e9c94e013ef69c89239829b1","abstract_canon_sha256":"f9154acd0579293c52d4f793d3e34c95d8d6030ced3e4b90ada7cf28deae7e61"},"schema_version":"1.0"},"canonical_sha256":"dae10141f26df5e3554d1fc83e742c1cc422c2352a67803715944bbbece4adf5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:52.302056Z","signature_b64":"OtNv//wNL986eKCXCOaivrI7C7vDXXeV10CWpr+ToMkt4sH0/Kaq/HvrcJI5fAh9+w/jxSXRmAzVssFa+FMeCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dae10141f26df5e3554d1fc83e742c1cc422c2352a67803715944bbbece4adf5","last_reissued_at":"2026-05-17T23:42:52.301572Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:52.301572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.08503","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":"2026-05-17T23:42:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1qPLKI+CGTVVtg6TRrEuk/XbrQh8mcmTTbDKGt4k05P9pdaNz61ma46jvUTsdj1PMvVgv8/CTFq8oRXJBQKdCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:33:45.242700Z"},"content_sha256":"e2e7d2e8216f21c4cfdda5237a02a4728a538bcdb346eb8ca06fa14959de8e8c","schema_version":"1.0","event_id":"sha256:e2e7d2e8216f21c4cfdda5237a02a4728a538bcdb346eb8ca06fa14959de8e8c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:3LQQCQPSNX26GVKND7ED45BMDT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Time fractional diffusion equations: solution concepts, regularity and long-time behaviour","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rico Zacher","submitted_at":"2019-06-20T08:53:20Z","abstract_excerpt":"In this paper we give a survey of results on various analytical aspects of time fractional diffusion equations. We describe the approach via abstract Volterra equations and collect results on strong solutions in the $L_p$ sense. We further discuss the concept of weak solutions for equations with rough coefficients and give an account of recent developments towards a De Giorgi-Nash-Moser theory for such equations. The last part summarizes recent results on the long-time behaviour of solutions, which turns out to be significantly different from that in the heat equation case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.08503","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":"2026-05-17T23:42:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D+shXOctu/ZW2xvsuEohkVhlVenLkNnhOqcmZAhXdhHPKLU34U/qJhxJbfEkWkjKTC52TnuvU/W+l0yoGqe7Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:33:45.243377Z"},"content_sha256":"1d8edacf03def00f6a32184276213fafa096c184c4fa790e60e25e38ad414dbe","schema_version":"1.0","event_id":"sha256:1d8edacf03def00f6a32184276213fafa096c184c4fa790e60e25e38ad414dbe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3LQQCQPSNX26GVKND7ED45BMDT/bundle.json","state_url":"https://pith.science/pith/3LQQCQPSNX26GVKND7ED45BMDT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3LQQCQPSNX26GVKND7ED45BMDT/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":"2026-05-26T05:33:45Z","links":{"resolver":"https://pith.science/pith/3LQQCQPSNX26GVKND7ED45BMDT","bundle":"https://pith.science/pith/3LQQCQPSNX26GVKND7ED45BMDT/bundle.json","state":"https://pith.science/pith/3LQQCQPSNX26GVKND7ED45BMDT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3LQQCQPSNX26GVKND7ED45BMDT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:3LQQCQPSNX26GVKND7ED45BMDT","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":"f9154acd0579293c52d4f793d3e34c95d8d6030ced3e4b90ada7cf28deae7e61","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T08:53:20Z","title_canon_sha256":"1b425810c1e31cbb63083cfe698c233bfb8d6fe3e9c94e013ef69c89239829b1"},"schema_version":"1.0","source":{"id":"1906.08503","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.08503","created_at":"2026-05-17T23:42:52Z"},{"alias_kind":"arxiv_version","alias_value":"1906.08503v1","created_at":"2026-05-17T23:42:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.08503","created_at":"2026-05-17T23:42:52Z"},{"alias_kind":"pith_short_12","alias_value":"3LQQCQPSNX26","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"3LQQCQPSNX26GVKN","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"3LQQCQPS","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:1d8edacf03def00f6a32184276213fafa096c184c4fa790e60e25e38ad414dbe","target":"graph","created_at":"2026-05-17T23:42:52Z","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":"In this paper we give a survey of results on various analytical aspects of time fractional diffusion equations. We describe the approach via abstract Volterra equations and collect results on strong solutions in the $L_p$ sense. We further discuss the concept of weak solutions for equations with rough coefficients and give an account of recent developments towards a De Giorgi-Nash-Moser theory for such equations. The last part summarizes recent results on the long-time behaviour of solutions, which turns out to be significantly different from that in the heat equation case.","authors_text":"Rico Zacher","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T08:53:20Z","title":"Time fractional diffusion equations: solution concepts, regularity and long-time behaviour"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.08503","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:e2e7d2e8216f21c4cfdda5237a02a4728a538bcdb346eb8ca06fa14959de8e8c","target":"record","created_at":"2026-05-17T23:42:52Z","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":"f9154acd0579293c52d4f793d3e34c95d8d6030ced3e4b90ada7cf28deae7e61","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T08:53:20Z","title_canon_sha256":"1b425810c1e31cbb63083cfe698c233bfb8d6fe3e9c94e013ef69c89239829b1"},"schema_version":"1.0","source":{"id":"1906.08503","kind":"arxiv","version":1}},"canonical_sha256":"dae10141f26df5e3554d1fc83e742c1cc422c2352a67803715944bbbece4adf5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dae10141f26df5e3554d1fc83e742c1cc422c2352a67803715944bbbece4adf5","first_computed_at":"2026-05-17T23:42:52.301572Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:52.301572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OtNv//wNL986eKCXCOaivrI7C7vDXXeV10CWpr+ToMkt4sH0/Kaq/HvrcJI5fAh9+w/jxSXRmAzVssFa+FMeCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:52.302056Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.08503","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e2e7d2e8216f21c4cfdda5237a02a4728a538bcdb346eb8ca06fa14959de8e8c","sha256:1d8edacf03def00f6a32184276213fafa096c184c4fa790e60e25e38ad414dbe"],"state_sha256":"d2610a9a12b7520ac5bbee94880181c4b9f217fb5c0841b3ac5f5804fa3f1f13"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aQyfKRiGTdpCWOnbP7VRAN6iO1RulR032N3sUgpLEn26eiIDEX6edA7xRvLDOZf3pb6AstPLG/Oxhh7hyY/yDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T05:33:45.246598Z","bundle_sha256":"75e2c3e0ee980cbbfecfd05f2010a12d671ab4c2458c5f6bc29d2ceae91cfe7a"}}