{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:LDFTENXYABUYB7F7DSCZBQ7ESI","short_pith_number":"pith:LDFTENXY","schema_version":"1.0","canonical_sha256":"58cb3236f8006980fcbf1c8590c3e49201800b6774ccf5716168aeb8c17cf0c6","source":{"kind":"arxiv","id":"1804.00560","version":1},"attestation_state":"computed","paper":{"title":"A blowup solution of a complex semi-linear heat equation with an irrational power","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Giao Ky Duong","submitted_at":"2018-03-30T16:57:30Z","abstract_excerpt":"In this paper, we consider the following semi-linear complex heat equation \\begin{eqnarray*} \\partial_t u = \\Delta u + u^p, u \\in \\mathbb{C} \\end{eqnarray*} in $\\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be non integer and even irrational. We construct 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 $a.$ Moreover, we also describe the asymptotics of the solution by the following final profiles: \\begin{eqnarray*} u(x,T) &\\sim & \\left[ \\frac{(p-1)^2 |x-a|^2}{ 8 p |\\ln|x-a||}\\right]^{-\\frac{1}"},"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":"1804.00560","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-03-30T16:57:30Z","cross_cats_sorted":[],"title_canon_sha256":"a4f64067bcb9dcbe5cc2a1eb3af4697593f9762890935b3a296e28facd80c894","abstract_canon_sha256":"62fea4a5c1fb87b8dc347adb3ba2b9f48a14f4f1d2caae54ec2ccc5e71b4a733"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:36.647914Z","signature_b64":"RLspCRnzKGcmZccIMkuFWLOTBGGRvOe/mHqSe9KVSocnOaI3ccLBYmC5jk9snsg6yUxJpvacd1sFUwh2N7OHBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58cb3236f8006980fcbf1c8590c3e49201800b6774ccf5716168aeb8c17cf0c6","last_reissued_at":"2026-05-18T00:19:36.647221Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:36.647221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A blowup solution of a complex semi-linear heat equation with an irrational power","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Giao Ky Duong","submitted_at":"2018-03-30T16:57:30Z","abstract_excerpt":"In this paper, we consider the following semi-linear complex heat equation \\begin{eqnarray*} \\partial_t u = \\Delta u + u^p, u \\in \\mathbb{C} \\end{eqnarray*} in $\\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be non integer and even irrational. We construct 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 $a.$ Moreover, we also describe the asymptotics of the solution by the following final profiles: \\begin{eqnarray*} u(x,T) &\\sim & \\left[ \\frac{(p-1)^2 |x-a|^2}{ 8 p |\\ln|x-a||}\\right]^{-\\frac{1}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.00560","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":"1804.00560","created_at":"2026-05-18T00:19:36.647347+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.00560v1","created_at":"2026-05-18T00:19:36.647347+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.00560","created_at":"2026-05-18T00:19:36.647347+00:00"},{"alias_kind":"pith_short_12","alias_value":"LDFTENXYABUY","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_16","alias_value":"LDFTENXYABUYB7F7","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_8","alias_value":"LDFTENXY","created_at":"2026-05-18T12:32:37.024351+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/LDFTENXYABUYB7F7DSCZBQ7ESI","json":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI.json","graph_json":"https://pith.science/api/pith-number/LDFTENXYABUYB7F7DSCZBQ7ESI/graph.json","events_json":"https://pith.science/api/pith-number/LDFTENXYABUYB7F7DSCZBQ7ESI/events.json","paper":"https://pith.science/paper/LDFTENXY"},"agent_actions":{"view_html":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI","download_json":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI.json","view_paper":"https://pith.science/paper/LDFTENXY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.00560&json=true","fetch_graph":"https://pith.science/api/pith-number/LDFTENXYABUYB7F7DSCZBQ7ESI/graph.json","fetch_events":"https://pith.science/api/pith-number/LDFTENXYABUYB7F7DSCZBQ7ESI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI/action/storage_attestation","attest_author":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI/action/author_attestation","sign_citation":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI/action/citation_signature","submit_replication":"https://pith.science/pith/LDFTENXYABUYB7F7DSCZBQ7ESI/action/replication_record"}},"created_at":"2026-05-18T00:19:36.647347+00:00","updated_at":"2026-05-18T00:19:36.647347+00:00"}