{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:KBQ2YC4EMPTZTW3VYHMOA4S2UT","short_pith_number":"pith:KBQ2YC4E","schema_version":"1.0","canonical_sha256":"5061ac0b8463e799db75c1d8e0725aa4c5a5ee19d4b183e3b4f24c6487dbcc9a","source":{"kind":"arxiv","id":"1211.4666","version":2},"attestation_state":"computed","paper":{"title":"Scattering theory for energy-supercritical Klein-Gordon equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Jiqiang Zheng","submitted_at":"2012-11-20T05:26:52Z","abstract_excerpt":"In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\\Delta u+u+|u|^2u=0$ in dimension $d\\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\\in L_t^\\infty(I; H^{s_c}_x(\\R^d)\\times H_x^{s_c-1}(\\R^d))$ with $s_c:=\\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schr\\\"odinger equation. However, the scaling invariance is broken"},"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":"1211.4666","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-20T05:26:52Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"bf9737ef0c70baa0a6425f4ecb8594e7cdafb28e6ee6952b6d4c169d831c480a","abstract_canon_sha256":"641dd3b92d2346bc701af9ab4ffead15960ea545c716d7472bf80929f9981191"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:35.427796Z","signature_b64":"Bwe2ipqeZjUYf9YkOfkH19THRF3RwPnoj9abFFGfviFrmlNtulgXjf2OA1dHx8PbfXfoY/LW8eReUOXqpnvbBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5061ac0b8463e799db75c1d8e0725aa4c5a5ee19d4b183e3b4f24c6487dbcc9a","last_reissued_at":"2026-05-18T00:49:35.427246Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:35.427246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering theory for energy-supercritical Klein-Gordon equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Jiqiang Zheng","submitted_at":"2012-11-20T05:26:52Z","abstract_excerpt":"In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\\Delta u+u+|u|^2u=0$ in dimension $d\\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\\in L_t^\\infty(I; H^{s_c}_x(\\R^d)\\times H_x^{s_c-1}(\\R^d))$ with $s_c:=\\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schr\\\"odinger equation. However, the scaling invariance is broken"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4666","kind":"arxiv","version":2},"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":"1211.4666","created_at":"2026-05-18T00:49:35.427333+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.4666v2","created_at":"2026-05-18T00:49:35.427333+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4666","created_at":"2026-05-18T00:49:35.427333+00:00"},{"alias_kind":"pith_short_12","alias_value":"KBQ2YC4EMPTZ","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"KBQ2YC4EMPTZTW3V","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"KBQ2YC4E","created_at":"2026-05-18T12:27:11.947152+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/KBQ2YC4EMPTZTW3VYHMOA4S2UT","json":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT.json","graph_json":"https://pith.science/api/pith-number/KBQ2YC4EMPTZTW3VYHMOA4S2UT/graph.json","events_json":"https://pith.science/api/pith-number/KBQ2YC4EMPTZTW3VYHMOA4S2UT/events.json","paper":"https://pith.science/paper/KBQ2YC4E"},"agent_actions":{"view_html":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT","download_json":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT.json","view_paper":"https://pith.science/paper/KBQ2YC4E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.4666&json=true","fetch_graph":"https://pith.science/api/pith-number/KBQ2YC4EMPTZTW3VYHMOA4S2UT/graph.json","fetch_events":"https://pith.science/api/pith-number/KBQ2YC4EMPTZTW3VYHMOA4S2UT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT/action/storage_attestation","attest_author":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT/action/author_attestation","sign_citation":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT/action/citation_signature","submit_replication":"https://pith.science/pith/KBQ2YC4EMPTZTW3VYHMOA4S2UT/action/replication_record"}},"created_at":"2026-05-18T00:49:35.427333+00:00","updated_at":"2026-05-18T00:49:35.427333+00:00"}