{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:5SLCMLZHPLGT2D2BSMZLDWODI6","short_pith_number":"pith:5SLCMLZH","schema_version":"1.0","canonical_sha256":"ec96262f277acd3d0f419332b1d9c347ab83b0f2ee47fd39732e44f8b0d06eaf","source":{"kind":"arxiv","id":"1201.4986","version":2},"attestation_state":"computed","paper":{"title":"Profiles of bounded radial solutions of the focusing, energy-critical wave equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Carlos Kenig, Frank Merle, Thomas Duyckaerts","submitted_at":"2012-01-24T14:33:59Z","abstract_excerpt":"In this paper we consider global and non-global bounded radial solutions of the focusing energy-critical wave equation in space dimension 3. We show that any of these solutions decouples, along a sequence of times that goes to the maximal time of existence, as a sum of modulated stationary solutions, a free radiation term and a term going to 0 in the energy space. In the case where there is only one stationary profile, we show that this expansion holds asymptotically without restriction to a subsequence."},"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":"1201.4986","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-24T14:33:59Z","cross_cats_sorted":[],"title_canon_sha256":"aa14b01160d835368bd569a67c7c912d0b161453532d35948ee4842a9928363b","abstract_canon_sha256":"f333b53ae621faefc4836144fc6f78180f72f9dc90c6c22ace06f52797c99d02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:31.889712Z","signature_b64":"PexWfPBDIUPfTTwF5aNGXv3NTxTX8TdOTP2O6OzgQI8a6NP/Z7K02K9p278HzaRHKImNGk1zDGpKVzG2nnSfDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec96262f277acd3d0f419332b1d9c347ab83b0f2ee47fd39732e44f8b0d06eaf","last_reissued_at":"2026-05-18T01:36:31.889210Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:31.889210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Profiles of bounded radial solutions of the focusing, energy-critical wave equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Carlos Kenig, Frank Merle, Thomas Duyckaerts","submitted_at":"2012-01-24T14:33:59Z","abstract_excerpt":"In this paper we consider global and non-global bounded radial solutions of the focusing energy-critical wave equation in space dimension 3. We show that any of these solutions decouples, along a sequence of times that goes to the maximal time of existence, as a sum of modulated stationary solutions, a free radiation term and a term going to 0 in the energy space. In the case where there is only one stationary profile, we show that this expansion holds asymptotically without restriction to a subsequence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.4986","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":"1201.4986","created_at":"2026-05-18T01:36:31.889296+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.4986v2","created_at":"2026-05-18T01:36:31.889296+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.4986","created_at":"2026-05-18T01:36:31.889296+00:00"},{"alias_kind":"pith_short_12","alias_value":"5SLCMLZHPLGT","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_16","alias_value":"5SLCMLZHPLGT2D2B","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_8","alias_value":"5SLCMLZH","created_at":"2026-05-18T12:26:56.085431+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/5SLCMLZHPLGT2D2BSMZLDWODI6","json":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6.json","graph_json":"https://pith.science/api/pith-number/5SLCMLZHPLGT2D2BSMZLDWODI6/graph.json","events_json":"https://pith.science/api/pith-number/5SLCMLZHPLGT2D2BSMZLDWODI6/events.json","paper":"https://pith.science/paper/5SLCMLZH"},"agent_actions":{"view_html":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6","download_json":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6.json","view_paper":"https://pith.science/paper/5SLCMLZH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.4986&json=true","fetch_graph":"https://pith.science/api/pith-number/5SLCMLZHPLGT2D2BSMZLDWODI6/graph.json","fetch_events":"https://pith.science/api/pith-number/5SLCMLZHPLGT2D2BSMZLDWODI6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6/action/storage_attestation","attest_author":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6/action/author_attestation","sign_citation":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6/action/citation_signature","submit_replication":"https://pith.science/pith/5SLCMLZHPLGT2D2BSMZLDWODI6/action/replication_record"}},"created_at":"2026-05-18T01:36:31.889296+00:00","updated_at":"2026-05-18T01:36:31.889296+00:00"}