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Pinsky","submitted_at":"2004-12-13T15:42:34Z","abstract_excerpt":"Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=\\Delta u -u^p  \\text{in} (R^n-\\{0\\})\\times(0,\\infty); & u(x,0)=g(x)\\ge0 \\text{in} R^n-\\{0\\}; &u\\ge0 \\text{in} (R^n-\\{0\\})\\times[0,\\infty). $ We prove that if $p\\ge\\frac n{n-2}$, then the solution to \\eqref{abstract} is unique for each $g$. On the other hand, if $p<\\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data."},"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":"math/0412241","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2004-12-13T15:42:34Z","cross_cats_sorted":[],"title_canon_sha256":"0fbd02971b7ce5b8b781eb1e5bc3659dd9570972fe43c98dcee7b128f1ed45d5","abstract_canon_sha256":"d2c9ef339d03e0b74c2b738c0c4c9d11535912abfdd3c993b1b86d479c0e4c7a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:23.939898Z","signature_b64":"xAcPiNrFCn/93pCMPqUMfJu5SgL2BxUU8fI8a4844e1ZJ3GLSI24vYqjosuQA4VYVboNOjkYlcFYTaH9J3TCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb377c1230838be4afee7f4efcd7548c0bd2a87c1c530ec5749c478d0f4cc8c3","last_reissued_at":"2026-05-18T01:05:23.939098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:23.939098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem for $u_t=\\Delta u-u^p$ in a punctured space","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ross G. Pinsky","submitted_at":"2004-12-13T15:42:34Z","abstract_excerpt":"Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=\\Delta u -u^p  \\text{in} (R^n-\\{0\\})\\times(0,\\infty); & u(x,0)=g(x)\\ge0 \\text{in} R^n-\\{0\\}; &u\\ge0 \\text{in} (R^n-\\{0\\})\\times[0,\\infty). $ We prove that if $p\\ge\\frac n{n-2}$, then the solution to \\eqref{abstract} is unique for each $g$. On the other hand, if $p<\\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0412241","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":"math/0412241","created_at":"2026-05-18T01:05:23.939223+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0412241v1","created_at":"2026-05-18T01:05:23.939223+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0412241","created_at":"2026-05-18T01:05:23.939223+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZM3XYERQQOF6","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZM3XYERQQOF6JL7O","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZM3XYERQ","created_at":"2026-05-18T12:25:52.687210+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/ZM3XYERQQOF6JL7OP5HPZV2URQ","json":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ.json","graph_json":"https://pith.science/api/pith-number/ZM3XYERQQOF6JL7OP5HPZV2URQ/graph.json","events_json":"https://pith.science/api/pith-number/ZM3XYERQQOF6JL7OP5HPZV2URQ/events.json","paper":"https://pith.science/paper/ZM3XYERQ"},"agent_actions":{"view_html":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ","download_json":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ.json","view_paper":"https://pith.science/paper/ZM3XYERQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0412241&json=true","fetch_graph":"https://pith.science/api/pith-number/ZM3XYERQQOF6JL7OP5HPZV2URQ/graph.json","fetch_events":"https://pith.science/api/pith-number/ZM3XYERQQOF6JL7OP5HPZV2URQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ/action/storage_attestation","attest_author":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ/action/author_attestation","sign_citation":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ/action/citation_signature","submit_replication":"https://pith.science/pith/ZM3XYERQQOF6JL7OP5HPZV2URQ/action/replication_record"}},"created_at":"2026-05-18T01:05:23.939223+00:00","updated_at":"2026-05-18T01:05:23.939223+00:00"}