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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"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1511.01885","created_at":"2026-05-18T01:27:39.564025+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.01885v1","created_at":"2026-05-18T01:27:39.564025+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.01885","created_at":"2026-05-18T01:27:39.564025+00:00"},{"alias_kind":"pith_short_12","alias_value":"H4YG3MXQZ3DL","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"H4YG3MXQZ3DLZ6LM","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"H4YG3MXQ","created_at":"2026-05-18T12:29:22.688609+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/H4YG3MXQZ3DLZ6LMNL6X2EGNAU","json":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU.json","graph_json":"https://pith.science/api/pith-number/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/graph.json","events_json":"https://pith.science/api/pith-number/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/events.json","paper":"https://pith.science/paper/H4YG3MXQ"},"agent_actions":{"view_html":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU","download_json":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU.json","view_paper":"https://pith.science/paper/H4YG3MXQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.01885&json=true","fetch_graph":"https://pith.science/api/pith-number/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/graph.json","fetch_events":"https://pith.science/api/pith-number/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/action/storage_attestation","attest_author":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/action/author_attestation","sign_citation":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/action/citation_signature","submit_replication":"https://pith.science/pith/H4YG3MXQZ3DLZ6LMNL6X2EGNAU/action/replication_record"}},"created_at":"2026-05-18T01:27:39.564025+00:00","updated_at":"2026-05-18T01:27:39.564025+00:00"}