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Besides obtaining sufficient conditions on the blowup in finite time and global existence of the solution, we establish Morawetz estimates and spacetime bounds for the global solution based o"},"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":"1904.09702","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2019-04-22T02:46:21Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"a0cc72acd1484718fe649e25acf4262aac62037c2d8c72d2c7e4e2cb2dcb5c7a","abstract_canon_sha256":"43af2b112fad72d5c273e566cc0fdb0cc235ff48d7ab34015f29216b742cbda1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:03.619560Z","signature_b64":"xBMJowz4EmcCXz5F/YNawvIkpQrQUbbv0PkuIG8XOi2/EWpp3ERofizPqZhzrASAdjWaTcZsOBFi5MIuSx4aDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4fe7633985445a8a15078d27ff544137e32634916a65d8f02e0fd61c3f443511","last_reissued_at":"2026-05-17T23:48:03.619109Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:03.619109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Morawetz estimates and spacetime bounds for quasilinear Schr\\\"{o}dinger equations with critical Sobolev exponent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Xianfa Song","submitted_at":"2019-04-22T02:46:21Z","abstract_excerpt":"In this paper, we study the following Cauchy problem \\begin{equation*} \\left\\{ \\begin{array}{lll} iu_t=\\Delta u + 2uh'(|u|^2)\\Delta h(|u|^2) + F(|u|^2)u\\mp A[h(|u|^2]^{2^*-1} h'(|u|^2)u,\\ x\\in \\mathbb{R}^N, \\ t>0\\\\ u(x,0)=u_0(x), \\quad x\\in \\mathbb{R}^N. \\end{array}\\right. \\end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, $h(s)\\geq 0$ and $h'(s)\\geq 0$ for $s\\geq 0$, $N\\geq 3$, $A>0$. Besides obtaining sufficient conditions on the blowup in finite time and global existence of the solution, we establish Morawetz estimates and spacetime bounds for the global solution based o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.09702","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":"1904.09702","created_at":"2026-05-17T23:48:03.619175+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.09702v1","created_at":"2026-05-17T23:48:03.619175+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.09702","created_at":"2026-05-17T23:48:03.619175+00:00"},{"alias_kind":"pith_short_12","alias_value":"J7TWGOMFIRNI","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"J7TWGOMFIRNIUFIH","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"J7TWGOMF","created_at":"2026-05-18T12:33:18.533446+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/J7TWGOMFIRNIUFIHRUT76VCBG7","json":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7.json","graph_json":"https://pith.science/api/pith-number/J7TWGOMFIRNIUFIHRUT76VCBG7/graph.json","events_json":"https://pith.science/api/pith-number/J7TWGOMFIRNIUFIHRUT76VCBG7/events.json","paper":"https://pith.science/paper/J7TWGOMF"},"agent_actions":{"view_html":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7","download_json":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7.json","view_paper":"https://pith.science/paper/J7TWGOMF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.09702&json=true","fetch_graph":"https://pith.science/api/pith-number/J7TWGOMFIRNIUFIHRUT76VCBG7/graph.json","fetch_events":"https://pith.science/api/pith-number/J7TWGOMFIRNIUFIHRUT76VCBG7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7/action/storage_attestation","attest_author":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7/action/author_attestation","sign_citation":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7/action/citation_signature","submit_replication":"https://pith.science/pith/J7TWGOMFIRNIUFIHRUT76VCBG7/action/replication_record"}},"created_at":"2026-05-17T23:48:03.619175+00:00","updated_at":"2026-05-17T23:48:03.619175+00:00"}