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Firstly, we prove the existence of singular solution $u$ of the above equation that is trapped in between self-similar solutions of the form of $t^{-\\alpha} f_i(t^{-\\beta}x)$, $i=1,2$, with initial value $u_0$ satisfying $A_1|x|^{-\\gamma}\\le u_0\\le A_2|x|^{-\\gamma}$ for some constants $A_2>A_1>0$ and $\\frac{2}{1-m}<\\gamma<\\frac{n-2}{m}$, where $\\beta:=\\frac{1}{2-\\gamma(1-m)}$, $"},"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":"1508.01980","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-09T03:09:21Z","cross_cats_sorted":[],"title_canon_sha256":"58e2d195cc6fdf5bbc9d723551b81ba60ee31e9318f6e6252a5d49a8b08b7703","abstract_canon_sha256":"38f3fc0e636b05b925dd1b49c0d40345f33797fbbdf755948f1eb3f890a6194c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:37.405780Z","signature_b64":"kPb8KdkoM0TM5VeRPl310D5gHBcmBILDWLVTV2OlJcg5OBn/21en4oFSXVCGwfdUUEYu0XPFfNRs5Ico9AKbBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc65f6bc24e7b26376194371e08ef7b3f2a069d9bd01797e5b3794cb4d59c9d7","last_reissued_at":"2026-05-18T01:35:37.405045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:37.405045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic large time behavior of singular solutions of the fast diffusion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui, Soojung Kim","submitted_at":"2015-08-09T03:09:21Z","abstract_excerpt":"We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=\\Delta u^m$ in $({\\mathbb R}^n\\setminus\\{0\\})\\times(0,\\infty)$ in the subcritical case $0<m<\\frac{n-2}{n}$, $n\\ge3$. Firstly, we prove the existence of singular solution $u$ of the above equation that is trapped in between self-similar solutions of the form of $t^{-\\alpha} f_i(t^{-\\beta}x)$, $i=1,2$, with initial value $u_0$ satisfying $A_1|x|^{-\\gamma}\\le u_0\\le A_2|x|^{-\\gamma}$ for some constants $A_2>A_1>0$ and $\\frac{2}{1-m}<\\gamma<\\frac{n-2}{m}$, where $\\beta:=\\frac{1}{2-\\gamma(1-m)}$, $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.01980","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":"1508.01980","created_at":"2026-05-18T01:35:37.405167+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.01980v1","created_at":"2026-05-18T01:35:37.405167+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.01980","created_at":"2026-05-18T01:35:37.405167+00:00"},{"alias_kind":"pith_short_12","alias_value":"7RS7NPBE46ZG","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"7RS7NPBE46ZGG5QZ","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"7RS7NPBE","created_at":"2026-05-18T12:29:10.953037+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/7RS7NPBE46ZGG5QZINY6BDXXWP","json":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP.json","graph_json":"https://pith.science/api/pith-number/7RS7NPBE46ZGG5QZINY6BDXXWP/graph.json","events_json":"https://pith.science/api/pith-number/7RS7NPBE46ZGG5QZINY6BDXXWP/events.json","paper":"https://pith.science/paper/7RS7NPBE"},"agent_actions":{"view_html":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP","download_json":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP.json","view_paper":"https://pith.science/paper/7RS7NPBE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.01980&json=true","fetch_graph":"https://pith.science/api/pith-number/7RS7NPBE46ZGG5QZINY6BDXXWP/graph.json","fetch_events":"https://pith.science/api/pith-number/7RS7NPBE46ZGG5QZINY6BDXXWP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP/action/storage_attestation","attest_author":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP/action/author_attestation","sign_citation":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP/action/citation_signature","submit_replication":"https://pith.science/pith/7RS7NPBE46ZGG5QZINY6BDXXWP/action/replication_record"}},"created_at":"2026-05-18T01:35:37.405167+00:00","updated_at":"2026-05-18T01:35:37.405167+00:00"}