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We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = \\Delta u + u^{\\frac{n+2}{n-2}} \\inn \\Omega\\times (0,\\infty), \\quad u =0 \\onn \\pp\\Omega\\times (0,\\infty). $$\n  Let $G(x,y)$ be the Dirichlet Green's function of $-\\Delta$ in $\\Omega$ and $H(x,y)$ its regular part. Let $q_j\\in \\Omega$, $j=1,\\ldots,k$, be points such that the matrix $$\n  \\left [ \\begin{matrix} H(q_1, q_1) & -G(q_1,q_2) &\\cdots & -G(q_1, q_k) -G(q_1,q_2) & H(q_2,q_2) & -G(q_2,q_3) \\cdots & -G(q_3,q_k) \\vdots & & \\ddots& \\vdots -G(q_1,q_k) "},"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":"1604.07117","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-25T03:20:45Z","cross_cats_sorted":[],"title_canon_sha256":"ba7fb81f29d167e394a8da822c344a3b25c7da14737b2cdca065464caad3850e","abstract_canon_sha256":"8cc514f551bd4dbe358deb6ff673436e0c5acd942b925d4d2cf945bf837fd561"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:22.970300Z","signature_b64":"KzmyoU5V3w7pRnZjXMdpkPkL3e3SeFQ1pptJJglG054MBvb/67g/Ew5DOndiqv04r3HFwCsM/aF4gwSr8a83AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44a4352946ca5a9d5daaa451eecf86068a9ecb8cae319b6b42f8e8cebb8b76a4","last_reissued_at":"2026-05-18T01:16:22.969696Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:22.969696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Green's function and infinite-time bubbling in the critical nonlinear heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Carmen Cortazar, Manuel del Pino, Monica Musso","submitted_at":"2016-04-25T03:20:45Z","abstract_excerpt":"Let $\\Omega$ be a smooth bounded domain in $\\R^n$, $n\\ge 5$. We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = \\Delta u + u^{\\frac{n+2}{n-2}} \\inn \\Omega\\times (0,\\infty), \\quad u =0 \\onn \\pp\\Omega\\times (0,\\infty). $$\n  Let $G(x,y)$ be the Dirichlet Green's function of $-\\Delta$ in $\\Omega$ and $H(x,y)$ its regular part. Let $q_j\\in \\Omega$, $j=1,\\ldots,k$, be points such that the matrix $$\n  \\left [ \\begin{matrix} H(q_1, q_1) & -G(q_1,q_2) &\\cdots & -G(q_1, q_k) -G(q_1,q_2) & H(q_2,q_2) & -G(q_2,q_3) \\cdots & -G(q_3,q_k) \\vdots & & \\ddots& \\vdots -G(q_1,q_k) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07117","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":"1604.07117","created_at":"2026-05-18T01:16:22.969795+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.07117v1","created_at":"2026-05-18T01:16:22.969795+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07117","created_at":"2026-05-18T01:16:22.969795+00:00"},{"alias_kind":"pith_short_12","alias_value":"ISSDKKKGZJNJ","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"ISSDKKKGZJNJ2XNK","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"ISSDKKKG","created_at":"2026-05-18T12:30:22.444734+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1907.06885","citing_title":"Construction of multi-bubble solutions for the energy-critical wave equation in dimension 5","ref_index":3,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2","json":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2.json","graph_json":"https://pith.science/api/pith-number/ISSDKKKGZJNJ2XNKURI65T4GA2/graph.json","events_json":"https://pith.science/api/pith-number/ISSDKKKGZJNJ2XNKURI65T4GA2/events.json","paper":"https://pith.science/paper/ISSDKKKG"},"agent_actions":{"view_html":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2","download_json":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2.json","view_paper":"https://pith.science/paper/ISSDKKKG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.07117&json=true","fetch_graph":"https://pith.science/api/pith-number/ISSDKKKGZJNJ2XNKURI65T4GA2/graph.json","fetch_events":"https://pith.science/api/pith-number/ISSDKKKGZJNJ2XNKURI65T4GA2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2/action/storage_attestation","attest_author":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2/action/author_attestation","sign_citation":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2/action/citation_signature","submit_replication":"https://pith.science/pith/ISSDKKKGZJNJ2XNKURI65T4GA2/action/replication_record"}},"created_at":"2026-05-18T01:16:22.969795+00:00","updated_at":"2026-05-18T01:16:22.969795+00:00"}