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Letting $Q$ be the ground state solution of $-Q+\\Delta Q+(|\\cdot|^{-3}\\ast|Q|^{2})Q=0 $ in $ \\mathbb{R}^{5}$, we prove that if $u_{0}\\in H^{1}$ satisfying $M(u_0) E(u_0)<M(Q) E(Q)$ and\n  $\\|\\nabla u_{0}\\|_{2}\\|u_{0}\\|_{2} >\\|\\nabla Q\\|_{2}\\|Q\\|_{2} ,$ then the corresponding solution $u(t)$ either blows up in finite forward time, or exists globally for pos"},"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":"1101.2053","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-01-11T06:54:35Z","cross_cats_sorted":[],"title_canon_sha256":"22676ff1ace07c520c86adc24677e99e680ddb07c8aaaf3b976e2b66fd36c1a3","abstract_canon_sha256":"6d7d952ccc27e0204fb3d3d524a3b2b6b005aa17ce9805f7c715974ad125fc35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:50.891848Z","signature_b64":"0ll31136zdQse3O4YLAsn/2kMMUMoTWyk/SWNc+Mf1nXmWTPtJf0UkSKWzkXGTztMfxNmwCgL3+Gcw7GCjEzBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce1b28d26b80d207c3cb5b26035361eb1cae6f35662b98c5968e48c358ef5c74","last_reissued_at":"2026-05-18T04:31:50.891461Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:50.891461Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Divergent solutions to the 5D Hartree Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daomin Cao, Qing Guo","submitted_at":"2011-01-11T06:54:35Z","abstract_excerpt":"We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\\Delta u+(|\\cdot|^{-3}\\ast|u|^{2})u=0$ in $\\mathbb{R}^{5}$ with the initial data in $H^1$, and study the divergent property of infinite-variance and nonradial solutions. Letting $Q$ be the ground state solution of $-Q+\\Delta Q+(|\\cdot|^{-3}\\ast|Q|^{2})Q=0 $ in $ \\mathbb{R}^{5}$, we prove that if $u_{0}\\in H^{1}$ satisfying $M(u_0) E(u_0)<M(Q) E(Q)$ and\n  $\\|\\nabla u_{0}\\|_{2}\\|u_{0}\\|_{2} >\\|\\nabla Q\\|_{2}\\|Q\\|_{2} ,$ then the corresponding solution $u(t)$ either blows up in finite forward time, or exists globally for pos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2053","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":"1101.2053","created_at":"2026-05-18T04:31:50.891521+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.2053v1","created_at":"2026-05-18T04:31:50.891521+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.2053","created_at":"2026-05-18T04:31:50.891521+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZYNSRUTLQDJA","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZYNSRUTLQDJAPQ6L","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZYNSRUTL","created_at":"2026-05-18T12:26:50.516681+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/ZYNSRUTLQDJAPQ6LLMTAGU3B5M","json":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M.json","graph_json":"https://pith.science/api/pith-number/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/graph.json","events_json":"https://pith.science/api/pith-number/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/events.json","paper":"https://pith.science/paper/ZYNSRUTL"},"agent_actions":{"view_html":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M","download_json":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M.json","view_paper":"https://pith.science/paper/ZYNSRUTL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.2053&json=true","fetch_graph":"https://pith.science/api/pith-number/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/graph.json","fetch_events":"https://pith.science/api/pith-number/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/action/storage_attestation","attest_author":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/action/author_attestation","sign_citation":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/action/citation_signature","submit_replication":"https://pith.science/pith/ZYNSRUTLQDJAPQ6LLMTAGU3B5M/action/replication_record"}},"created_at":"2026-05-18T04:31:50.891521+00:00","updated_at":"2026-05-18T04:31:50.891521+00:00"}