{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:V3XN4SUDBOE62PBQKIEQD3NJDD","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":"06b56c73118b0b9e6f2f145477caf7bc0a65347f69bbf77b581bddf0c54505fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-19T03:18:43Z","title_canon_sha256":"20bbbbaa8e5f0904fb015f41ebef2c10b5bd0824ba28e5d766e0a60de7c44671"},"schema_version":"1.0","source":{"id":"1809.06994","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.06994","created_at":"2026-05-17T23:51:25Z"},{"alias_kind":"arxiv_version","alias_value":"1809.06994v1","created_at":"2026-05-17T23:51:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.06994","created_at":"2026-05-17T23:51:25Z"},{"alias_kind":"pith_short_12","alias_value":"V3XN4SUDBOE6","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V3XN4SUDBOE62PBQ","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V3XN4SUD","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:649c5acaf402e3d98bc3cc37bdd5535395deb4339b01044291283326205e9748","target":"graph","created_at":"2026-05-17T23:51:25Z","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"},"paper":{"abstract_excerpt":"We consider the Cauchy problem of the semilinear wave equation with a damping term \\begin{align*}\n  u_{tt} - \\Delta u + c(t,x) u_t = |u|^p, \\quad (t,x)\\in (0,\\infty)\\times \\mathbb{R}^N,\\quad\n  u(0,x) = \\varepsilon u_0(x), \\ u_t(0,x) = \\varepsilon u_1(x), \\quad x\\in \\mathbb{R}^N, \\end{align*} where $p>1$ and the coefficient of the damping term has the form \\begin{align*}\n  c(t,x) = a_0 (1+|x|^2)^{-\\alpha/2} (1+t)^{-\\beta} \\end{align*} with some $a_0 > 0$, $\\alpha < 0$, $\\beta \\in (-1, 1]$. In particular, we mainly consider the cases $ \\alpha < 0, \\beta =0$ or $\\alpha < 0, \\beta = 1$, which impl","authors_text":"Kenji Nishihara, Motohiro Sobajima, Yuta Wakasugi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-19T03:18:43Z","title":"Critical exponent for the semilinear wave equations with a damping increasing in the far field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.06994","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":"sha256:1daac487d4e29c8b342509ca6254a0ccce57f5a1d95b2eb5b8e18818dd88141b","target":"record","created_at":"2026-05-17T23:51:25Z","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":"06b56c73118b0b9e6f2f145477caf7bc0a65347f69bbf77b581bddf0c54505fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-19T03:18:43Z","title_canon_sha256":"20bbbbaa8e5f0904fb015f41ebef2c10b5bd0824ba28e5d766e0a60de7c44671"},"schema_version":"1.0","source":{"id":"1809.06994","kind":"arxiv","version":1}},"canonical_sha256":"aeeede4a830b89ed3c30520901eda918e47a0b78fa955e515a6b4e3a2fcdd52a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aeeede4a830b89ed3c30520901eda918e47a0b78fa955e515a6b4e3a2fcdd52a","first_computed_at":"2026-05-17T23:51:25.924054Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:25.924054Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jvL61tTr4MSRtmITUKRMJoUw7wn77ArW24lqfBN+xEChxPbzV6ADBZnZa3NHlve1rvv4KTEsfh5p0b3x9fdVBA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:25.924729Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.06994","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1daac487d4e29c8b342509ca6254a0ccce57f5a1d95b2eb5b8e18818dd88141b","sha256:649c5acaf402e3d98bc3cc37bdd5535395deb4339b01044291283326205e9748"],"state_sha256":"ed44c0857e191228d73369f1e43a23eeb83394441b001276a7dc463adbb63b5f"}