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Torebek, Mohamed Majdoub","submitted_at":"2026-06-25T03:04:39Z","abstract_excerpt":"We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy--H\\'enon-type weight \\[\n  u_t + (-\\Delta)^s u = |x|^{-\\gamma}\\,|u|^p,\n  \\qquad (t,x)\\in(0,\\infty)\\times\\mathbb{R}^N,\n  \\qquad u(0,x)=\\varepsilon\\,u_0(x), \\] where $0<s<1$, $0\\le\\gamma<\\min(2s,N)$, $p>1$ and $u_0\\in L^1\\cap L^\\infty$ with $\\int_{\\mathbb{R}^N}u_0(x)\\,dx>0$. Setting \\[\n  p_F \\;:=\\; 1+\\frac{2s-\\gamma}{N}, \\] we prove that the lifespan $T_\\varepsilon$ obeys, for every sufficiently small $\\varepsilon>0$, \\[\n  T_\\varepsilon \\;\\approx\\;\n  \\begin{cases}\n  \\varepsilon^{-\\,\\beta^{-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":"2606.26555","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-25T03:04:39Z","cross_cats_sorted":[],"title_canon_sha256":"cb276a9ad900474e58586e6f4032a41cc9cd9a7d35c4be11f3b84713f921b437","abstract_canon_sha256":"3cfd8ec5f4558d8486187e9998667236b4e4a1f669415664dba34f9de7c38b63"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T01:15:34.482929Z","signature_b64":"XMYRUKrCO4jInK1uurpBei4Y0bTiewIHKggU9AaBgTeKwsFsZNJs0Ul+kK2ZZBrs0B2bQO2A98aj5teS4X8BDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e50515ab29092463eafef4eef73443cc0ef5e8222dd46e8fe9c4944b4b916d89","last_reissued_at":"2026-06-26T01:15:34.482524Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T01:15:34.482524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy-H\\'enon Type Parabolic Equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Berikbol T. Torebek, Mohamed Majdoub","submitted_at":"2026-06-25T03:04:39Z","abstract_excerpt":"We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy--H\\'enon-type weight \\[\n  u_t + (-\\Delta)^s u = |x|^{-\\gamma}\\,|u|^p,\n  \\qquad (t,x)\\in(0,\\infty)\\times\\mathbb{R}^N,\n  \\qquad u(0,x)=\\varepsilon\\,u_0(x), \\] where $0<s<1$, $0\\le\\gamma<\\min(2s,N)$, $p>1$ and $u_0\\in L^1\\cap L^\\infty$ with $\\int_{\\mathbb{R}^N}u_0(x)\\,dx>0$. Setting \\[\n  p_F \\;:=\\; 1+\\frac{2s-\\gamma}{N}, \\] we prove that the lifespan $T_\\varepsilon$ obeys, for every sufficiently small $\\varepsilon>0$, \\[\n  T_\\varepsilon \\;\\approx\\;\n  \\begin{cases}\n  \\varepsilon^{-\\,\\beta^{-1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.26555","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.26555/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.26555","created_at":"2026-06-26T01:15:34.482583+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.26555v1","created_at":"2026-06-26T01:15:34.482583+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.26555","created_at":"2026-06-26T01:15:34.482583+00:00"},{"alias_kind":"pith_short_12","alias_value":"4UCRLKZJBESG","created_at":"2026-06-26T01:15:34.482583+00:00"},{"alias_kind":"pith_short_16","alias_value":"4UCRLKZJBESGH2X6","created_at":"2026-06-26T01:15:34.482583+00:00"},{"alias_kind":"pith_short_8","alias_value":"4UCRLKZJ","created_at":"2026-06-26T01:15:34.482583+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/4UCRLKZJBESGH2X66TXPONCDZQ","json":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ.json","graph_json":"https://pith.science/api/pith-number/4UCRLKZJBESGH2X66TXPONCDZQ/graph.json","events_json":"https://pith.science/api/pith-number/4UCRLKZJBESGH2X66TXPONCDZQ/events.json","paper":"https://pith.science/paper/4UCRLKZJ"},"agent_actions":{"view_html":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ","download_json":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ.json","view_paper":"https://pith.science/paper/4UCRLKZJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.26555&json=true","fetch_graph":"https://pith.science/api/pith-number/4UCRLKZJBESGH2X66TXPONCDZQ/graph.json","fetch_events":"https://pith.science/api/pith-number/4UCRLKZJBESGH2X66TXPONCDZQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ/action/storage_attestation","attest_author":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ/action/author_attestation","sign_citation":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ/action/citation_signature","submit_replication":"https://pith.science/pith/4UCRLKZJBESGH2X66TXPONCDZQ/action/replication_record"}},"created_at":"2026-06-26T01:15:34.482583+00:00","updated_at":"2026-06-26T01:15:34.482583+00:00"}