{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:H4YG3MXQZ3DLZ6LMNL6X2EGNAU","short_pith_number":"pith:H4YG3MXQ","canonical_record":{"source":{"id":"1511.01885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-05T20:37:25Z","cross_cats_sorted":[],"title_canon_sha256":"40935989ca842cf1c87c5f4e1e15093716de215579ff77e9dbd86db3966a2df0","abstract_canon_sha256":"998321417a8128ff5de0efbae7ca6e0fd05fcf7d33865295a7ce3275e0ee81b7"},"schema_version":"1.0"},"canonical_sha256":"3f306db2f0cec6bcf96c6afd7d10cd0516a0be081d3a062b33e7f97ad867237f","source":{"kind":"arxiv","id":"1511.01885","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.01885","created_at":"2026-05-18T01:27:39Z"},{"alias_kind":"arxiv_version","alias_value":"1511.01885v1","created_at":"2026-05-18T01:27:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.01885","created_at":"2026-05-18T01:27:39Z"},{"alias_kind":"pith_short_12","alias_value":"H4YG3MXQZ3DL","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"H4YG3MXQZ3DLZ6LM","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"H4YG3MXQ","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:H4YG3MXQZ3DLZ6LMNL6X2EGNAU","target":"record","payload":{"canonical_record":{"source":{"id":"1511.01885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-05T20:37:25Z","cross_cats_sorted":[],"title_canon_sha256":"40935989ca842cf1c87c5f4e1e15093716de215579ff77e9dbd86db3966a2df0","abstract_canon_sha256":"998321417a8128ff5de0efbae7ca6e0fd05fcf7d33865295a7ce3275e0ee81b7"},"schema_version":"1.0"},"canonical_sha256":"3f306db2f0cec6bcf96c6afd7d10cd0516a0be081d3a062b33e7f97ad867237f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:39.564544Z","signature_b64":"wjdmbpFlKwWExwsvEyK0KBLyGhNfcL22A/HqDGhiDhoU1pAuw+e4IgCp638xvxL+BQmvbbmT0RDX76e85CqUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f306db2f0cec6bcf96c6afd7d10cd0516a0be081d3a062b33e7f97ad867237f","last_reissued_at":"2026-05-18T01:27:39.563933Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:39.563933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.01885","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":"2026-05-18T01:27:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LrR17TZLCPNoHjIKB09Mt4F4FAYQheq2J8Donl6BiwOuQPBIzPsku1YkR3zTyKpsb/4OrKQOTLbctlDPf7JTCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T04:53:06.422961Z"},"content_sha256":"dc05345fb5de078e785734880910ef7297bf6c5f8640897fec8d620bf4600245","schema_version":"1.0","event_id":"sha256:dc05345fb5de078e785734880910ef7297bf6c5f8640897fec8d620bf4600245"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:H4YG3MXQZ3DLZ6LMNL6X2EGNAU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Johannes Lankeit","submitted_at":"2015-11-05T20:37:25Z","abstract_excerpt":"We prove convergence of positive solutions to \\[ u_t = u\\Delta u + u\\int_{\\Omega} |\\nabla u|^2, \\qquad u\\rvert_{\\partial\\Omega} =0, \\qquad u(\\cdot,0)=u_0 \\] in a bounded domain $\\Omega\\subset \\mathbb{R}^n$, $n\\ge 1$, with smooth boundary in the case of $\\int_\\Omega u_0=1$ and identify the $W_0^{1,2}(\\Omega)$-limit of $u(t)$ as $t\\to \\infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\\int_\\Omega u_0<1$ and $\\int_\\Omega u_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively.\n\n  The proof is base"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.01885","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":"2026-05-18T01:27:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PFHEsYiWQQtjP8tdJSNI1P4+PynLc2pnvZkQ44r5a7qPu8pDj9uwdnA0IDgyEtC0AuKUfuP5oxcexjro/uRqBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T04:53:06.423310Z"},"content_sha256":"66915c184858d9aa2d5bcea403af07603273fc2e9053a41fd03a7301b55779a8","schema_version":"1.0","event_id":"sha256:66915c184858d9aa2d5bcea403af07603273fc2e9053a41fd03a7301b55779a8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/bundle.json","state_url":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/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":"2026-06-22T04:53:06Z","links":{"resolver":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU","bundle":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/bundle.json","state":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:H4YG3MXQZ3DLZ6LMNL6X2EGNAU","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":"998321417a8128ff5de0efbae7ca6e0fd05fcf7d33865295a7ce3275e0ee81b7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-05T20:37:25Z","title_canon_sha256":"40935989ca842cf1c87c5f4e1e15093716de215579ff77e9dbd86db3966a2df0"},"schema_version":"1.0","source":{"id":"1511.01885","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.01885","created_at":"2026-05-18T01:27:39Z"},{"alias_kind":"arxiv_version","alias_value":"1511.01885v1","created_at":"2026-05-18T01:27:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.01885","created_at":"2026-05-18T01:27:39Z"},{"alias_kind":"pith_short_12","alias_value":"H4YG3MXQZ3DL","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"H4YG3MXQZ3DLZ6LM","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"H4YG3MXQ","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:66915c184858d9aa2d5bcea403af07603273fc2e9053a41fd03a7301b55779a8","target":"graph","created_at":"2026-05-18T01:27:39Z","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 prove convergence of positive solutions to \\[ u_t = u\\Delta u + u\\int_{\\Omega} |\\nabla u|^2, \\qquad u\\rvert_{\\partial\\Omega} =0, \\qquad u(\\cdot,0)=u_0 \\] in a bounded domain $\\Omega\\subset \\mathbb{R}^n$, $n\\ge 1$, with smooth boundary in the case of $\\int_\\Omega u_0=1$ and identify the $W_0^{1,2}(\\Omega)$-limit of $u(t)$ as $t\\to \\infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\\int_\\Omega u_0<1$ and $\\int_\\Omega u_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively.\n\n  The proof is base","authors_text":"Johannes Lankeit","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-05T20:37:25Z","title":"Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.01885","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:dc05345fb5de078e785734880910ef7297bf6c5f8640897fec8d620bf4600245","target":"record","created_at":"2026-05-18T01:27:39Z","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":"998321417a8128ff5de0efbae7ca6e0fd05fcf7d33865295a7ce3275e0ee81b7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-05T20:37:25Z","title_canon_sha256":"40935989ca842cf1c87c5f4e1e15093716de215579ff77e9dbd86db3966a2df0"},"schema_version":"1.0","source":{"id":"1511.01885","kind":"arxiv","version":1}},"canonical_sha256":"3f306db2f0cec6bcf96c6afd7d10cd0516a0be081d3a062b33e7f97ad867237f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f306db2f0cec6bcf96c6afd7d10cd0516a0be081d3a062b33e7f97ad867237f","first_computed_at":"2026-05-18T01:27:39.563933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:27:39.563933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wjdmbpFlKwWExwsvEyK0KBLyGhNfcL22A/HqDGhiDhoU1pAuw+e4IgCp638xvxL+BQmvbbmT0RDX76e85CqUAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:27:39.564544Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.01885","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc05345fb5de078e785734880910ef7297bf6c5f8640897fec8d620bf4600245","sha256:66915c184858d9aa2d5bcea403af07603273fc2e9053a41fd03a7301b55779a8"],"state_sha256":"c6260c4a01d0fdeaf160fbb23da5d4cf8fe624b2a49d7b6fb0d8066ffaf6ba1f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MThQ8c9w8OQkxDAkbYiJK0QtiY+JXnxp69fNQowxPf0n9c6jWzwdKkAmKdxb4yxxY8SSO8tF5Dor0blF5GFvDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T04:53:06.425230Z","bundle_sha256":"fa56d57e873a0e17e983fad1e07d54e83f1dd9bd77474c9d6e3e0d662e8f9452"}}