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Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum \\cite{DP2}, \\cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=\\Delta\\log u$ in $\\R^2\\times (0,T)$, $u(x,0)=u_0(x)$ in $\\R"},"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":"0910.5045","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-10-27T05:50:54Z","cross_cats_sorted":[],"title_canon_sha256":"1e9d1d4741d9764e449c6656bd3ad8eb5cc6225469bd1fbe2f2f85fd1ef5c501","abstract_canon_sha256":"ee97f5bbaf9cc8720ff07719d709b503d8db0a27279daf50e2b52273fc9754d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:10.989479Z","signature_b64":"5OukmiaVOH/l+qqvYPhMs/KOhf6HSFe3PluXJfyX4Fu5ar+jaKFg6HjB3kN7Y5hH30q8HTSd9nEvilDxPv7OCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee024679e40434bcc45daf7c1d1c570dc6e858762430bbd6c2e50296ec643950","last_reissued_at":"2026-05-18T04:21:10.988870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:10.988870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Collapsing behaviour of a singular diffusion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui","submitted_at":"2009-10-27T05:50:54Z","abstract_excerpt":"Let $0\\le u_0(x)\\in L^1(\\R^2)\\cap L^{\\infty}(\\R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|\\ge r_1$ and is monotone decreasing for all $|x|\\ge r_1$ for some constant $r_1>0$ and ${ess}\\inf_{\\2{B}_{r_1}(0)}u_0\\ge{ess} \\sup_{\\R^2\\setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum \\cite{DP2}, \\cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=\\Delta\\log u$ in $\\R^2\\times (0,T)$, $u(x,0)=u_0(x)$ in $\\R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.5045","kind":"arxiv","version":2},"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":"0910.5045","created_at":"2026-05-18T04:21:10.988957+00:00"},{"alias_kind":"arxiv_version","alias_value":"0910.5045v2","created_at":"2026-05-18T04:21:10.988957+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.5045","created_at":"2026-05-18T04:21:10.988957+00:00"},{"alias_kind":"pith_short_12","alias_value":"5YBEM6PEAQ2L","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"5YBEM6PEAQ2LZRC5","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"5YBEM6PE","created_at":"2026-05-18T12:25:58.837520+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/5YBEM6PEAQ2LZRC5V56B2HCXBX","json":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX.json","graph_json":"https://pith.science/api/pith-number/5YBEM6PEAQ2LZRC5V56B2HCXBX/graph.json","events_json":"https://pith.science/api/pith-number/5YBEM6PEAQ2LZRC5V56B2HCXBX/events.json","paper":"https://pith.science/paper/5YBEM6PE"},"agent_actions":{"view_html":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX","download_json":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX.json","view_paper":"https://pith.science/paper/5YBEM6PE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0910.5045&json=true","fetch_graph":"https://pith.science/api/pith-number/5YBEM6PEAQ2LZRC5V56B2HCXBX/graph.json","fetch_events":"https://pith.science/api/pith-number/5YBEM6PEAQ2LZRC5V56B2HCXBX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX/action/storage_attestation","attest_author":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX/action/author_attestation","sign_citation":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX/action/citation_signature","submit_replication":"https://pith.science/pith/5YBEM6PEAQ2LZRC5V56B2HCXBX/action/replication_record"}},"created_at":"2026-05-18T04:21:10.988957+00:00","updated_at":"2026-05-18T04:21:10.988957+00:00"}