{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:JOXWJQRYEQPFUL6TVSZF3ORBBI","short_pith_number":"pith:JOXWJQRY","canonical_record":{"source":{"id":"1904.10534","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-21T19:05:03Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"82c5f0ad75053608a0ff98f170dfc1ed9c3033766b825496fa122ea00555bf50","abstract_canon_sha256":"750f8cbbed82cab37314666756c4707f5c31f9fe12a4271c874114e37a3952eb"},"schema_version":"1.0"},"canonical_sha256":"4baf64c238241e5a2fd3acb25dba210a028236da80b948c735fb76a4706eee07","source":{"kind":"arxiv","id":"1904.10534","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.10534","created_at":"2026-05-17T23:47:53Z"},{"alias_kind":"arxiv_version","alias_value":"1904.10534v1","created_at":"2026-05-17T23:47:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.10534","created_at":"2026-05-17T23:47:53Z"},{"alias_kind":"pith_short_12","alias_value":"JOXWJQRYEQPF","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"JOXWJQRYEQPFUL6T","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"JOXWJQRY","created_at":"2026-05-18T12:33:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:JOXWJQRYEQPFUL6TVSZF3ORBBI","target":"record","payload":{"canonical_record":{"source":{"id":"1904.10534","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-21T19:05:03Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"82c5f0ad75053608a0ff98f170dfc1ed9c3033766b825496fa122ea00555bf50","abstract_canon_sha256":"750f8cbbed82cab37314666756c4707f5c31f9fe12a4271c874114e37a3952eb"},"schema_version":"1.0"},"canonical_sha256":"4baf64c238241e5a2fd3acb25dba210a028236da80b948c735fb76a4706eee07","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:53.476347Z","signature_b64":"xjnaELEFAI14VwWh5gQ6ljYbejh3ohE8s0k2Uow4MOWPhSwcFKNJTMjnWuuKQBX8iGSypY91eDFEDztrHPetCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4baf64c238241e5a2fd3acb25dba210a028236da80b948c735fb76a4706eee07","last_reissued_at":"2026-05-17T23:47:53.475785Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:53.475785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.10534","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":"2026-05-17T23:47:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fe7lMwWFXiQACGuBclQOFQOk49csFboJOzmUS2e7I3BAjE0JyxDkxQdkKODdFPj5AM50X8o2VbAoe2EvkS3lAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T10:19:41.431281Z"},"content_sha256":"1d7ab800d34cc7040fbd3b71c549726c748f52448fbb557188fae8753d55c6cd","schema_version":"1.0","event_id":"sha256:1d7ab800d34cc7040fbd3b71c549726c748f52448fbb557188fae8753d55c6cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:JOXWJQRYEQPFUL6TVSZF3ORBBI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global existence and uniqueness of the solution to a nonlinear parabolic equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Alexander G. Ramm","submitted_at":"2019-04-21T19:05:03Z","abstract_excerpt":"Consider the equation $$ u'(t)-\\Delta u+|u|^\\rho u=0, \\quad u(0)=u_0(x), (1), $$ where $ u':=\\frac {du}{dt}$, $ \\rho=const >0, $ $x\\in \\mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\\|u_0\\|\\:=\\sup_{x\\in \\mathbb{R}^3, t\\in \\mathbb{R}_+} |u(x,t)|.$$ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate $$\\|u(x,t)\\|<c, $$ where $c>0$ does not depend on $x,t$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.10534","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":"2026-05-17T23:47:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yg1vUENXYoWTLCzzE+vOSi/ixNu+JCxtgO2XiXxkOsnDTlZFtXcKC7i6pt1uy+1tUtAQzYcS4jONIHBJ4IfcAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T10:19:41.431631Z"},"content_sha256":"ab2def8443b99c9ba668cfcdbc87966e1aa110f5fea166cb60ec36ab31481cc3","schema_version":"1.0","event_id":"sha256:ab2def8443b99c9ba668cfcdbc87966e1aa110f5fea166cb60ec36ab31481cc3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JOXWJQRYEQPFUL6TVSZF3ORBBI/bundle.json","state_url":"https://pith.science/pith/JOXWJQRYEQPFUL6TVSZF3ORBBI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JOXWJQRYEQPFUL6TVSZF3ORBBI/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":"2026-06-27T10:19:41Z","links":{"resolver":"https://pith.science/pith/JOXWJQRYEQPFUL6TVSZF3ORBBI","bundle":"https://pith.science/pith/JOXWJQRYEQPFUL6TVSZF3ORBBI/bundle.json","state":"https://pith.science/pith/JOXWJQRYEQPFUL6TVSZF3ORBBI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JOXWJQRYEQPFUL6TVSZF3ORBBI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:JOXWJQRYEQPFUL6TVSZF3ORBBI","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":"750f8cbbed82cab37314666756c4707f5c31f9fe12a4271c874114e37a3952eb","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-21T19:05:03Z","title_canon_sha256":"82c5f0ad75053608a0ff98f170dfc1ed9c3033766b825496fa122ea00555bf50"},"schema_version":"1.0","source":{"id":"1904.10534","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.10534","created_at":"2026-05-17T23:47:53Z"},{"alias_kind":"arxiv_version","alias_value":"1904.10534v1","created_at":"2026-05-17T23:47:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.10534","created_at":"2026-05-17T23:47:53Z"},{"alias_kind":"pith_short_12","alias_value":"JOXWJQRYEQPF","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"JOXWJQRYEQPFUL6T","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"JOXWJQRY","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:ab2def8443b99c9ba668cfcdbc87966e1aa110f5fea166cb60ec36ab31481cc3","target":"graph","created_at":"2026-05-17T23:47:53Z","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":"Consider the equation $$ u'(t)-\\Delta u+|u|^\\rho u=0, \\quad u(0)=u_0(x), (1), $$ where $ u':=\\frac {du}{dt}$, $ \\rho=const >0, $ $x\\in \\mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\\|u_0\\|\\:=\\sup_{x\\in \\mathbb{R}^3, t\\in \\mathbb{R}_+} |u(x,t)|.$$ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate $$\\|u(x,t)\\|<c, $$ where $c>0$ does not depend on $x,t$.","authors_text":"Alexander G. Ramm","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-21T19:05:03Z","title":"Global existence and uniqueness of the solution to a nonlinear parabolic equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.10534","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":"sha256:1d7ab800d34cc7040fbd3b71c549726c748f52448fbb557188fae8753d55c6cd","target":"record","created_at":"2026-05-17T23:47:53Z","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":"750f8cbbed82cab37314666756c4707f5c31f9fe12a4271c874114e37a3952eb","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-21T19:05:03Z","title_canon_sha256":"82c5f0ad75053608a0ff98f170dfc1ed9c3033766b825496fa122ea00555bf50"},"schema_version":"1.0","source":{"id":"1904.10534","kind":"arxiv","version":1}},"canonical_sha256":"4baf64c238241e5a2fd3acb25dba210a028236da80b948c735fb76a4706eee07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4baf64c238241e5a2fd3acb25dba210a028236da80b948c735fb76a4706eee07","first_computed_at":"2026-05-17T23:47:53.475785Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:47:53.475785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xjnaELEFAI14VwWh5gQ6ljYbejh3ohE8s0k2Uow4MOWPhSwcFKNJTMjnWuuKQBX8iGSypY91eDFEDztrHPetCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:47:53.476347Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.10534","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d7ab800d34cc7040fbd3b71c549726c748f52448fbb557188fae8753d55c6cd","sha256:ab2def8443b99c9ba668cfcdbc87966e1aa110f5fea166cb60ec36ab31481cc3"],"state_sha256":"8b7dd3d9f5b84e5fef97938e115380bd2670e8e942829e8cd927d2ca52cb7dd4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+h1ZtBwEyZEViK7lT3KBhN0hSMHc7q8hzHu+icfy2NzZNr4fsZ6Vji8lWosRWxqfJ97epqqwcCHVw3sPxmF1CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T10:19:41.433461Z","bundle_sha256":"495f5d7468d4c0aed72c1e6e01fa805b2b54c9e59e539d6ec58e45da989f1d16"}}