{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:CI2D43EHQUD76LDGGG7TJIZAXT","short_pith_number":"pith:CI2D43EH","schema_version":"1.0","canonical_sha256":"12343e6c878507ff2c6631bf34a320bcedc5edab4392d4b6703cb34e90adf458","source":{"kind":"arxiv","id":"1111.6671","version":1},"attestation_state":"computed","paper":{"title":"The dynamics of the 3D radial NLS with the combined terms","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Lifeng Zhao","submitted_at":"2011-11-29T02:37:43Z","abstract_excerpt":"In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\\\"{o}dinger equation (NLS) with the combined terms\niu_t + \\Delta u = -|u|^4u + |u|^2u \\tag{CNLS}\nin the energy space $H^1(\\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \\Delta u = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \\cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \\cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition wi"},"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":"1111.6671","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2011-11-29T02:37:43Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"00943ad1ceba07d30bce72ef87d757ccc0691ec2ad79c142eab75be42ea2c2ab","abstract_canon_sha256":"450e20d65acdb55a115b068ad138cee5c4771c0c5dcb912baaca16dde58e192b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:52.439372Z","signature_b64":"hCPQxtThJhOiVbYoATjmSzvtZIT6aD9lROTKEUXYWW8M4e4VWJvTamqAVY0/uPhTTIL8H6cv27TaxxYD4yRUDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12343e6c878507ff2c6631bf34a320bcedc5edab4392d4b6703cb34e90adf458","last_reissued_at":"2026-05-18T03:27:52.438845Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:52.438845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The dynamics of the 3D radial NLS with the combined terms","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Guixiang Xu, Lifeng Zhao","submitted_at":"2011-11-29T02:37:43Z","abstract_excerpt":"In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\\\"{o}dinger equation (NLS) with the combined terms\niu_t + \\Delta u = -|u|^4u + |u|^2u \\tag{CNLS}\nin the energy space $H^1(\\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \\Delta u = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \\cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \\cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6671","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":"1111.6671","created_at":"2026-05-18T03:27:52.438933+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.6671v1","created_at":"2026-05-18T03:27:52.438933+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6671","created_at":"2026-05-18T03:27:52.438933+00:00"},{"alias_kind":"pith_short_12","alias_value":"CI2D43EHQUD7","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"CI2D43EHQUD76LDG","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"CI2D43EH","created_at":"2026-05-18T12:26:26.731475+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/CI2D43EHQUD76LDGGG7TJIZAXT","json":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT.json","graph_json":"https://pith.science/api/pith-number/CI2D43EHQUD76LDGGG7TJIZAXT/graph.json","events_json":"https://pith.science/api/pith-number/CI2D43EHQUD76LDGGG7TJIZAXT/events.json","paper":"https://pith.science/paper/CI2D43EH"},"agent_actions":{"view_html":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT","download_json":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT.json","view_paper":"https://pith.science/paper/CI2D43EH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.6671&json=true","fetch_graph":"https://pith.science/api/pith-number/CI2D43EHQUD76LDGGG7TJIZAXT/graph.json","fetch_events":"https://pith.science/api/pith-number/CI2D43EHQUD76LDGGG7TJIZAXT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT/action/storage_attestation","attest_author":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT/action/author_attestation","sign_citation":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT/action/citation_signature","submit_replication":"https://pith.science/pith/CI2D43EHQUD76LDGGG7TJIZAXT/action/replication_record"}},"created_at":"2026-05-18T03:27:52.438933+00:00","updated_at":"2026-05-18T03:27:52.438933+00:00"}