{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:535NXWUJTSAUJKEKPREUDJPC2K","short_pith_number":"pith:535NXWUJ","canonical_record":{"source":{"id":"2401.04115","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-12-22T07:41:13Z","cross_cats_sorted":[],"title_canon_sha256":"11866beee0a5f803efd51c72da4085e9ba9bb068151544cbc4e24263c5790c2e","abstract_canon_sha256":"0f033541bb2e6affa166ac1eace9804ff5c2c466fdd7f028e81ec76e0de1e3d7"},"schema_version":"1.0"},"canonical_sha256":"eefadbda899c8144a88a7c4941a5e2d297b7937a09d0a46dc92e8ce8cb0ad4ca","source":{"kind":"arxiv","id":"2401.04115","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.04115","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"arxiv_version","alias_value":"2401.04115v3","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.04115","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"pith_short_12","alias_value":"535NXWUJTSAU","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"pith_short_16","alias_value":"535NXWUJTSAUJKEK","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"pith_short_8","alias_value":"535NXWUJ","created_at":"2026-06-24T01:14:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:535NXWUJTSAUJKEKPREUDJPC2K","target":"record","payload":{"canonical_record":{"source":{"id":"2401.04115","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-12-22T07:41:13Z","cross_cats_sorted":[],"title_canon_sha256":"11866beee0a5f803efd51c72da4085e9ba9bb068151544cbc4e24263c5790c2e","abstract_canon_sha256":"0f033541bb2e6affa166ac1eace9804ff5c2c466fdd7f028e81ec76e0de1e3d7"},"schema_version":"1.0"},"canonical_sha256":"eefadbda899c8144a88a7c4941a5e2d297b7937a09d0a46dc92e8ce8cb0ad4ca","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:14:18.367546Z","signature_b64":"uF1BhfRUhVSIMqkDEbnvlCl/fGaDOIi8+8+1D4DPYdjrYy5q6/aiESOfs4xZw9Q58TF+OQiFQYm6s/rMouCrDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eefadbda899c8144a88a7c4941a5e2d297b7937a09d0a46dc92e8ce8cb0ad4ca","last_reissued_at":"2026-06-24T01:14:18.367077Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:14:18.367077Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2401.04115","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-24T01:14:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e18LnM1/XEMGnM1ejIjM+SMa/kk9jc7C0DAHLfgkiSLKVOOBMp+nyxVrYyVa3p6mxmNpzlLVR5phbLxa53MJBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T12:19:22.305671Z"},"content_sha256":"d477d3e83edb37bd3d681f82ff35df36851d30fb443893f2eca1a8db4a4fec75","schema_version":"1.0","event_id":"sha256:d477d3e83edb37bd3d681f82ff35df36851d30fb443893f2eca1a8db4a4fec75"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:535NXWUJTSAUJKEKPREUDJPC2K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Soliton resolution for the energy critical damped wave equations in the radial case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jingyuan Gu, Lifeng Zhao","submitted_at":"2023-12-22T07:41:13Z","abstract_excerpt":"We consider energy-critical damped wave equation \\begin{equation*} \\partial_{tt}u-\\Delta u+\\alpha \\partial_t u=\\left|u\\right|^{\\frac{4}{D-2}}u \\end{equation*} with radial initial data in dimensions $D\\geq 4$. The equation has a nontrivial radial stationary solution $W$, called the ground state, which is unique up to sign and scale. We prove that any bounded energy norm solution behaves asymptotically as a superposition of the modulated ground states and a radiation term. In the global case, particularly, the solution converges to a pure multi-bubble due to the damping effect."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.04115","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2401.04115/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-24T01:14:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0YLhuUjZmIr33lbYSVfVADzMFuEewzMHyGUGuo/7CPM+XosUx1hopDWwpupkcy0EySfiKscfQR7hTOFXXgPaBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T12:19:22.306041Z"},"content_sha256":"964bd2664006f96562c77126aa6bb1d0ca2acbbb399c0b97a74a56dcd41c77af","schema_version":"1.0","event_id":"sha256:964bd2664006f96562c77126aa6bb1d0ca2acbbb399c0b97a74a56dcd41c77af"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/535NXWUJTSAUJKEKPREUDJPC2K/bundle.json","state_url":"https://pith.science/pith/535NXWUJTSAUJKEKPREUDJPC2K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/535NXWUJTSAUJKEKPREUDJPC2K/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T12:19:22Z","links":{"resolver":"https://pith.science/pith/535NXWUJTSAUJKEKPREUDJPC2K","bundle":"https://pith.science/pith/535NXWUJTSAUJKEKPREUDJPC2K/bundle.json","state":"https://pith.science/pith/535NXWUJTSAUJKEKPREUDJPC2K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/535NXWUJTSAUJKEKPREUDJPC2K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:535NXWUJTSAUJKEKPREUDJPC2K","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0f033541bb2e6affa166ac1eace9804ff5c2c466fdd7f028e81ec76e0de1e3d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-12-22T07:41:13Z","title_canon_sha256":"11866beee0a5f803efd51c72da4085e9ba9bb068151544cbc4e24263c5790c2e"},"schema_version":"1.0","source":{"id":"2401.04115","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.04115","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"arxiv_version","alias_value":"2401.04115v3","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.04115","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"pith_short_12","alias_value":"535NXWUJTSAU","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"pith_short_16","alias_value":"535NXWUJTSAUJKEK","created_at":"2026-06-24T01:14:18Z"},{"alias_kind":"pith_short_8","alias_value":"535NXWUJ","created_at":"2026-06-24T01:14:18Z"}],"graph_snapshots":[{"event_id":"sha256:964bd2664006f96562c77126aa6bb1d0ca2acbbb399c0b97a74a56dcd41c77af","target":"graph","created_at":"2026-06-24T01:14:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2401.04115/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider energy-critical damped wave equation \\begin{equation*} \\partial_{tt}u-\\Delta u+\\alpha \\partial_t u=\\left|u\\right|^{\\frac{4}{D-2}}u \\end{equation*} with radial initial data in dimensions $D\\geq 4$. The equation has a nontrivial radial stationary solution $W$, called the ground state, which is unique up to sign and scale. We prove that any bounded energy norm solution behaves asymptotically as a superposition of the modulated ground states and a radiation term. In the global case, particularly, the solution converges to a pure multi-bubble due to the damping effect.","authors_text":"Jingyuan Gu, Lifeng Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-12-22T07:41:13Z","title":"Soliton resolution for the energy critical damped wave equations in the radial case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.04115","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d477d3e83edb37bd3d681f82ff35df36851d30fb443893f2eca1a8db4a4fec75","target":"record","created_at":"2026-06-24T01:14:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0f033541bb2e6affa166ac1eace9804ff5c2c466fdd7f028e81ec76e0de1e3d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-12-22T07:41:13Z","title_canon_sha256":"11866beee0a5f803efd51c72da4085e9ba9bb068151544cbc4e24263c5790c2e"},"schema_version":"1.0","source":{"id":"2401.04115","kind":"arxiv","version":3}},"canonical_sha256":"eefadbda899c8144a88a7c4941a5e2d297b7937a09d0a46dc92e8ce8cb0ad4ca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eefadbda899c8144a88a7c4941a5e2d297b7937a09d0a46dc92e8ce8cb0ad4ca","first_computed_at":"2026-06-24T01:14:18.367077Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-24T01:14:18.367077Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uF1BhfRUhVSIMqkDEbnvlCl/fGaDOIi8+8+1D4DPYdjrYy5q6/aiESOfs4xZw9Q58TF+OQiFQYm6s/rMouCrDw==","signature_status":"signed_v1","signed_at":"2026-06-24T01:14:18.367546Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.04115","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d477d3e83edb37bd3d681f82ff35df36851d30fb443893f2eca1a8db4a4fec75","sha256:964bd2664006f96562c77126aa6bb1d0ca2acbbb399c0b97a74a56dcd41c77af"],"state_sha256":"d07bb03616e8a232b2f183713bcc548362d0dcf0b82b1b22f444b2937adeadbf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zMzOY3GU7kvptGU7gheoibIzrFQbBrJhhgGbv5CISuSFAKepIu2EzWw7EFVe8lfpLNcHWnzgxtiLmrDFw6/VBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T12:19:22.308378Z","bundle_sha256":"0624f4bc374475d91f56d06b82408854c984d8c9cb631d1211705c51d05579d5"}}