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Suppose $u_0\\ge 0$ satisfies $u_0-\\psi_{\\lambda_0}\\in L^1(R^n)$ and $u_0(x)\\approx\\frac{2(n-2)}{\\beta}\\frac{\\log |x|}{|x|^2}$ as $|x|\\to\\infty$. We prove that the rescaled solution $\\widetilde{u}(x,t)=e^{2\\beta t}u(e^{\\beta t}x,t)$ of the maximal global solution $u$ of the equation $u_t=\\Delta\\log u$ in $R^n\\times (0,\\infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subse"},"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":"1111.5692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-24T08:34:45Z","cross_cats_sorted":[],"title_canon_sha256":"003fff006ffeed4f42475096953b0a727d8c0086ca27cc86fc07ab00785439e8","abstract_canon_sha256":"3039cbafcc1f097755e672bcac13c46bc7816dcea343d27df7d36ff3358ebdb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:42.549093Z","signature_b64":"AnmHF85Q4638kEn7aHOHUlXlk/tq2iSYGuOWdHeTB1/bp27V/J+hP855XcdmVL826iwRYVrXu9qQa+icqRCPDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15fb369633f2be57c98435c1a1051da09a1dcf3bd1f0d5d93b85e3e4ffd001b2","last_reissued_at":"2026-05-18T04:07:42.548486Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:42.548486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large time behaviour of higher dimensional 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":"2011-11-24T08:34:45Z","abstract_excerpt":"Let $n\\ge 3$ and $\\psi_{\\lambda_0}$ be the radially symmetric solution of $\\Delta\\log\\psi+2\\beta\\psi+\\beta x\\cdot\\nabla\\psi=0$ in $R^n$, $\\psi(0)=\\lambda_0$, for some constants $\\lambda_0>0$, $\\beta>0$. Suppose $u_0\\ge 0$ satisfies $u_0-\\psi_{\\lambda_0}\\in L^1(R^n)$ and $u_0(x)\\approx\\frac{2(n-2)}{\\beta}\\frac{\\log |x|}{|x|^2}$ as $|x|\\to\\infty$. We prove that the rescaled solution $\\widetilde{u}(x,t)=e^{2\\beta t}u(e^{\\beta t}x,t)$ of the maximal global solution $u$ of the equation $u_t=\\Delta\\log u$ in $R^n\\times (0,\\infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5692","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":"1111.5692","created_at":"2026-05-18T04:07:42.548602+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.5692v1","created_at":"2026-05-18T04:07:42.548602+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5692","created_at":"2026-05-18T04:07:42.548602+00:00"},{"alias_kind":"pith_short_12","alias_value":"CX5TNFRT6K7F","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"CX5TNFRT6K7FPSME","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"CX5TNFRT","created_at":"2026-05-18T12:26:26.731475+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/CX5TNFRT6K7FPSMEGXA2CBI5UC","json":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC.json","graph_json":"https://pith.science/api/pith-number/CX5TNFRT6K7FPSMEGXA2CBI5UC/graph.json","events_json":"https://pith.science/api/pith-number/CX5TNFRT6K7FPSMEGXA2CBI5UC/events.json","paper":"https://pith.science/paper/CX5TNFRT"},"agent_actions":{"view_html":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC","download_json":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC.json","view_paper":"https://pith.science/paper/CX5TNFRT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.5692&json=true","fetch_graph":"https://pith.science/api/pith-number/CX5TNFRT6K7FPSMEGXA2CBI5UC/graph.json","fetch_events":"https://pith.science/api/pith-number/CX5TNFRT6K7FPSMEGXA2CBI5UC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC/action/storage_attestation","attest_author":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC/action/author_attestation","sign_citation":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC/action/citation_signature","submit_replication":"https://pith.science/pith/CX5TNFRT6K7FPSMEGXA2CBI5UC/action/replication_record"}},"created_at":"2026-05-18T04:07:42.548602+00:00","updated_at":"2026-05-18T04:07:42.548602+00:00"}