{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:HZ5BLHSVRBQSWDXJ7DNKYOSY2R","short_pith_number":"pith:HZ5BLHSV","schema_version":"1.0","canonical_sha256":"3e7a159e5588612b0ee9f8daac3a58d44cef68b93f17a340582a82a06c84beae","source":{"kind":"arxiv","id":"1012.1915","version":4},"attestation_state":"computed","paper":{"title":"Extinction profile of the logarithmic diffusion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui, SungHoon Kim","submitted_at":"2010-12-09T03:23:55Z","abstract_excerpt":"Let $u$ be the solution of $u_t=\\Delta\\log u$ in $\\R^N\\times (0,T)$, N=3 or $N\\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\\le u_0\\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\\log (T-t)$, converges uniformly on $\\R^N$ to the rescaled Barenblatt solution $\\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\\to\\infty$. We also obtain convergence of the rescaled solution "},"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":"1012.1915","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-12-09T03:23:55Z","cross_cats_sorted":[],"title_canon_sha256":"100912ef593dfb2ae4621344b30d80809ae234ba1389242c43b3de8421ef29db","abstract_canon_sha256":"fd3ee06c5afd9d107de33694dc00c7f9f33621e812246c856a16eb0fe4d3f1c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:57.911035Z","signature_b64":"ajEoZ75MH0m7ycz/JiHncC4rl/Uu+CBu8tozLFnDlTGvmAkCpTpKa3xmfr7u5uBZabSGdHTa7kidwz5b3L3FAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e7a159e5588612b0ee9f8daac3a58d44cef68b93f17a340582a82a06c84beae","last_reissued_at":"2026-05-18T03:44:57.910442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:57.910442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extinction profile of the logarithmic diffusion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui, SungHoon Kim","submitted_at":"2010-12-09T03:23:55Z","abstract_excerpt":"Let $u$ be the solution of $u_t=\\Delta\\log u$ in $\\R^N\\times (0,T)$, N=3 or $N\\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\\le u_0\\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\\log (T-t)$, converges uniformly on $\\R^N$ to the rescaled Barenblatt solution $\\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\\to\\infty$. We also obtain convergence of the rescaled solution "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1915","kind":"arxiv","version":4},"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":"1012.1915","created_at":"2026-05-18T03:44:57.910555+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.1915v4","created_at":"2026-05-18T03:44:57.910555+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1915","created_at":"2026-05-18T03:44:57.910555+00:00"},{"alias_kind":"pith_short_12","alias_value":"HZ5BLHSVRBQS","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_16","alias_value":"HZ5BLHSVRBQSWDXJ","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_8","alias_value":"HZ5BLHSV","created_at":"2026-05-18T12:26:07.630475+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/HZ5BLHSVRBQSWDXJ7DNKYOSY2R","json":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R.json","graph_json":"https://pith.science/api/pith-number/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/graph.json","events_json":"https://pith.science/api/pith-number/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/events.json","paper":"https://pith.science/paper/HZ5BLHSV"},"agent_actions":{"view_html":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R","download_json":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R.json","view_paper":"https://pith.science/paper/HZ5BLHSV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.1915&json=true","fetch_graph":"https://pith.science/api/pith-number/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/graph.json","fetch_events":"https://pith.science/api/pith-number/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/action/storage_attestation","attest_author":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/action/author_attestation","sign_citation":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/action/citation_signature","submit_replication":"https://pith.science/pith/HZ5BLHSVRBQSWDXJ7DNKYOSY2R/action/replication_record"}},"created_at":"2026-05-18T03:44:57.910555+00:00","updated_at":"2026-05-18T03:44:57.910555+00:00"}