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We aim at constructing for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a$, with the following estimates for the final profile \\begin{eqnarray*} u(x,T) &\\sim & \\left[ \\frac{(p-1)^2 |x-a|^2}{ 8 p |\\ln|x-a||}\\right]^{-\\frac{1}{p-1}}, u_2(x,T) &\\sim & \\frac{2 p}{(p-1)^2} \\left[ \\frac{ (p-1)^2|x-"},"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":"1712.07183","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-19T20:02:40Z","cross_cats_sorted":[],"title_canon_sha256":"5118076f40745c42b23619b72fdfae3c2dde0751f0f4de36c828e5179d19d8d0","abstract_canon_sha256":"9fc0829d8e6217e6d8a366004b1901a3e05fd23471fe3a31c25cc24bb6b627d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:35.318922Z","signature_b64":"QTkRUAmtHN2iI7nlvUBkpt7O7YkqP9zVHJQVcz6datyJrvQLuQ7HyR58zt+aPspzXul5BYRbrcBM3BNVeZhOBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"75343b7c3f814ef96c74c5a529eb368f46f786e41c079867d3d16c510981cfca","last_reissued_at":"2026-05-18T00:27:35.318258Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:35.318258Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Profile for the imaginary part of a blowup solution for a complex-valued seminar heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Giao Ky Duong","submitted_at":"2017-12-19T20:02:40Z","abstract_excerpt":"In this paper, we consider the following complex-valued semilinear heat equation \\begin{eqnarray*} \\partial_t u = \\Delta u + u^p, u \\in \\mathbb{C}, \\end{eqnarray*} in the whole space $\\mathbb{R}^n$, where $ p \\in \\mathbb{N}, p \\geq 2$. We aim at constructing for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a$, with the following estimates for the final profile \\begin{eqnarray*} u(x,T) &\\sim & \\left[ \\frac{(p-1)^2 |x-a|^2}{ 8 p |\\ln|x-a||}\\right]^{-\\frac{1}{p-1}}, u_2(x,T) &\\sim & \\frac{2 p}{(p-1)^2} \\left[ \\frac{ (p-1)^2|x-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.07183","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":"1712.07183","created_at":"2026-05-18T00:27:35.318366+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.07183v1","created_at":"2026-05-18T00:27:35.318366+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.07183","created_at":"2026-05-18T00:27:35.318366+00:00"},{"alias_kind":"pith_short_12","alias_value":"OU2DW7B7QFHP","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"OU2DW7B7QFHPS3DU","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"OU2DW7B7","created_at":"2026-05-18T12:31:34.259226+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/OU2DW7B7QFHPS3DUYWSST2ZWR5","json":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5.json","graph_json":"https://pith.science/api/pith-number/OU2DW7B7QFHPS3DUYWSST2ZWR5/graph.json","events_json":"https://pith.science/api/pith-number/OU2DW7B7QFHPS3DUYWSST2ZWR5/events.json","paper":"https://pith.science/paper/OU2DW7B7"},"agent_actions":{"view_html":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5","download_json":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5.json","view_paper":"https://pith.science/paper/OU2DW7B7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.07183&json=true","fetch_graph":"https://pith.science/api/pith-number/OU2DW7B7QFHPS3DUYWSST2ZWR5/graph.json","fetch_events":"https://pith.science/api/pith-number/OU2DW7B7QFHPS3DUYWSST2ZWR5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5/action/storage_attestation","attest_author":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5/action/author_attestation","sign_citation":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5/action/citation_signature","submit_replication":"https://pith.science/pith/OU2DW7B7QFHPS3DUYWSST2ZWR5/action/replication_record"}},"created_at":"2026-05-18T00:27:35.318366+00:00","updated_at":"2026-05-18T00:27:35.318366+00:00"}