{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:YHT2F7KXZASHAVHMEICZ6ULJMG","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":"50146a46499262d0d8b4d966c03f505d46b903d9be1378653cef29c988f78e6c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-01-15T08:36:46Z","title_canon_sha256":"c1165c5c8e6dfe0995dff89182d291738ba59b9ca5ca0a18a8fc924846d21d73"},"schema_version":"1.0","source":{"id":"2101.06012","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.06012","created_at":"2026-07-05T02:07:09Z"},{"alias_kind":"arxiv_version","alias_value":"2101.06012v1","created_at":"2026-07-05T02:07:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.06012","created_at":"2026-07-05T02:07:09Z"},{"alias_kind":"pith_short_12","alias_value":"YHT2F7KXZASH","created_at":"2026-07-05T02:07:09Z"},{"alias_kind":"pith_short_16","alias_value":"YHT2F7KXZASHAVHM","created_at":"2026-07-05T02:07:09Z"},{"alias_kind":"pith_short_8","alias_value":"YHT2F7KX","created_at":"2026-07-05T02:07:09Z"}],"graph_snapshots":[{"event_id":"sha256:e47a69b2d43fb475026498fdc1694631fb937f6de613c867effc2b05c50b8797","target":"graph","created_at":"2026-07-05T02:07:09Z","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/2101.06012/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation\n  $$u_{tt}-\\left(a \\int_\\Omega |\\nabla u|^2 \\dif x +b\\right)\\Delta u = \\lambda u+ |u|^{p-1}u\n  ,$$\n  where $a$, $b>0$, $p>1$, $\\lambda \\in \\mathbb{R}$ and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of $a$, $b$, $\\lambda$ and the initial data for the following cases: (i) $1<p<3$, (ii) $p=3$ and $a>1/\\Lambda$, (iii) $p=3$, $a \\leq 1/\\Lambda$ and $\\lam <b\\lam_1$, (i","authors_text":"Jianyi Chen, Yimin Sun, Zhitao Zhang, Zonghu Xiu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-01-15T08:36:46Z","title":"Dynamics of nonlinear hyperbolic equations of Kirchhoff type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.06012","kind":"arxiv","version":1},"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":"sha256:48bdf8053659e87be373a2ef06852f586be7b265481668bbff5a988bcf27ac2c","target":"record","created_at":"2026-07-05T02:07:09Z","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":"50146a46499262d0d8b4d966c03f505d46b903d9be1378653cef29c988f78e6c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-01-15T08:36:46Z","title_canon_sha256":"c1165c5c8e6dfe0995dff89182d291738ba59b9ca5ca0a18a8fc924846d21d73"},"schema_version":"1.0","source":{"id":"2101.06012","kind":"arxiv","version":1}},"canonical_sha256":"c1e7a2fd57c8247054ec22059f516961badafebeddbd45493f0a0417ee68cc1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c1e7a2fd57c8247054ec22059f516961badafebeddbd45493f0a0417ee68cc1a","first_computed_at":"2026-07-05T02:07:09.024099Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:07:09.024099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"30jiwc4+XO4kXwFUbndHnmkIDpvPrWvfke1KD2wRRl52zSH0J3Qn0O077HWz9Z06wLSeeH8uOFiRbZZ1RXQxCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:07:09.024578Z","signed_message":"canonical_sha256_bytes"},"source_id":"2101.06012","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:48bdf8053659e87be373a2ef06852f586be7b265481668bbff5a988bcf27ac2c","sha256:e47a69b2d43fb475026498fdc1694631fb937f6de613c867effc2b05c50b8797"],"state_sha256":"61ead9f7c8626f73d52fe5aa7e745b35730115b65445f4912141934b44d025bb"}