{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:IUDALZSJZQPLWQ2B2UW5U5SEGW","short_pith_number":"pith:IUDALZSJ","canonical_record":{"source":{"id":"1807.06164","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-17T00:41:54Z","cross_cats_sorted":[],"title_canon_sha256":"1775b3fb121116581c7bff4deb1791be2d468ef2e9bd48a34f40e39422cc63c4","abstract_canon_sha256":"045e204720fb8ee786d518bcdb28efe0c60985e731d26404fbb732b379c35973"},"schema_version":"1.0"},"canonical_sha256":"450605e649cc1ebb4341d52dda7644358670c83ae0da451e7b607145463dc831","source":{"kind":"arxiv","id":"1807.06164","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.06164","created_at":"2026-05-18T00:10:33Z"},{"alias_kind":"arxiv_version","alias_value":"1807.06164v1","created_at":"2026-05-18T00:10:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.06164","created_at":"2026-05-18T00:10:33Z"},{"alias_kind":"pith_short_12","alias_value":"IUDALZSJZQPL","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"IUDALZSJZQPLWQ2B","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"IUDALZSJ","created_at":"2026-05-18T12:32:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:IUDALZSJZQPLWQ2B2UW5U5SEGW","target":"record","payload":{"canonical_record":{"source":{"id":"1807.06164","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-17T00:41:54Z","cross_cats_sorted":[],"title_canon_sha256":"1775b3fb121116581c7bff4deb1791be2d468ef2e9bd48a34f40e39422cc63c4","abstract_canon_sha256":"045e204720fb8ee786d518bcdb28efe0c60985e731d26404fbb732b379c35973"},"schema_version":"1.0"},"canonical_sha256":"450605e649cc1ebb4341d52dda7644358670c83ae0da451e7b607145463dc831","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:33.815644Z","signature_b64":"8Iw9fNLWTnkgtpGyCHAGBRuhv3eY9598zZ/yphuHq7ALLTSa7dbzBW+DvQ4YSbOvOVRHPflYsmZXe6eFeaMXDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"450605e649cc1ebb4341d52dda7644358670c83ae0da451e7b607145463dc831","last_reissued_at":"2026-05-18T00:10:33.815118Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:33.815118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.06164","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vfNYsvOmLzv8ICBR4wO5exrW49IvU9Q6brmKfqgv6baYmKfrnpI22NdypQU/oSvs0jO+Nl7G0lr/8kZvPRYdCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:28:30.411901Z"},"content_sha256":"79998749b649ca845d691552737e19f6f145cc82e1cc73071a02b9cd4ac0ec27","schema_version":"1.0","event_id":"sha256:79998749b649ca845d691552737e19f6f145cc82e1cc73071a02b9cd4ac0ec27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:IUDALZSJZQPLWQ2B2UW5U5SEGW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the blow-up for critical semilinear wave equations with damping in the scattering case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Borislav Yordanov, Kyouhei Wakasa","submitted_at":"2018-07-17T00:41:54Z","abstract_excerpt":"We consider the Cauchy problem for semilinear wave equations with variable coefficients and time-dependent scattering damping in $\\mathbf{R}^n$, where $n\\geq 2$. It is expected that the critical exponent will be Strauss' number $p_0(n)$, which is also the one for semilinear wave equations without damping terms. Lai and Takamura (2018) have obtained the blow-up part, together with the upper bound of lifespan, in the sub-critical case $p<p_0(n)$. In this paper, we extend their results to the critical case $p=p_0(n)$. The proof is based on Wakasa and Yordanov (2018), which concerns the blow-up an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06164","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zTs2DFdKB6ToYrCqabaEoj3wOvWIJCsybEvjWHVF62mUivhMdsUtg0cJOf/vh7O5v8L+OwOcnD2/5C6UXtwhBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T15:28:30.412575Z"},"content_sha256":"0b400ad050b3a1ce95f64c03d54b3edfee51502ff4a0b5cfbf8fbb2e0a33acfc","schema_version":"1.0","event_id":"sha256:0b400ad050b3a1ce95f64c03d54b3edfee51502ff4a0b5cfbf8fbb2e0a33acfc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IUDALZSJZQPLWQ2B2UW5U5SEGW/bundle.json","state_url":"https://pith.science/pith/IUDALZSJZQPLWQ2B2UW5U5SEGW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IUDALZSJZQPLWQ2B2UW5U5SEGW/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":"2026-06-22T15:28:30Z","links":{"resolver":"https://pith.science/pith/IUDALZSJZQPLWQ2B2UW5U5SEGW","bundle":"https://pith.science/pith/IUDALZSJZQPLWQ2B2UW5U5SEGW/bundle.json","state":"https://pith.science/pith/IUDALZSJZQPLWQ2B2UW5U5SEGW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IUDALZSJZQPLWQ2B2UW5U5SEGW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:IUDALZSJZQPLWQ2B2UW5U5SEGW","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":"045e204720fb8ee786d518bcdb28efe0c60985e731d26404fbb732b379c35973","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-17T00:41:54Z","title_canon_sha256":"1775b3fb121116581c7bff4deb1791be2d468ef2e9bd48a34f40e39422cc63c4"},"schema_version":"1.0","source":{"id":"1807.06164","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.06164","created_at":"2026-05-18T00:10:33Z"},{"alias_kind":"arxiv_version","alias_value":"1807.06164v1","created_at":"2026-05-18T00:10:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.06164","created_at":"2026-05-18T00:10:33Z"},{"alias_kind":"pith_short_12","alias_value":"IUDALZSJZQPL","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"IUDALZSJZQPLWQ2B","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"IUDALZSJ","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:0b400ad050b3a1ce95f64c03d54b3edfee51502ff4a0b5cfbf8fbb2e0a33acfc","target":"graph","created_at":"2026-05-18T00:10:33Z","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 for semilinear wave equations with variable coefficients and time-dependent scattering damping in $\\mathbf{R}^n$, where $n\\geq 2$. It is expected that the critical exponent will be Strauss' number $p_0(n)$, which is also the one for semilinear wave equations without damping terms. Lai and Takamura (2018) have obtained the blow-up part, together with the upper bound of lifespan, in the sub-critical case $p<p_0(n)$. In this paper, we extend their results to the critical case $p=p_0(n)$. The proof is based on Wakasa and Yordanov (2018), which concerns the blow-up an","authors_text":"Borislav Yordanov, Kyouhei Wakasa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-17T00:41:54Z","title":"On the blow-up for critical semilinear wave equations with damping in the scattering case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06164","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":"sha256:79998749b649ca845d691552737e19f6f145cc82e1cc73071a02b9cd4ac0ec27","target":"record","created_at":"2026-05-18T00:10:33Z","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":"045e204720fb8ee786d518bcdb28efe0c60985e731d26404fbb732b379c35973","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-17T00:41:54Z","title_canon_sha256":"1775b3fb121116581c7bff4deb1791be2d468ef2e9bd48a34f40e39422cc63c4"},"schema_version":"1.0","source":{"id":"1807.06164","kind":"arxiv","version":1}},"canonical_sha256":"450605e649cc1ebb4341d52dda7644358670c83ae0da451e7b607145463dc831","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"450605e649cc1ebb4341d52dda7644358670c83ae0da451e7b607145463dc831","first_computed_at":"2026-05-18T00:10:33.815118Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:33.815118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Iw9fNLWTnkgtpGyCHAGBRuhv3eY9598zZ/yphuHq7ALLTSa7dbzBW+DvQ4YSbOvOVRHPflYsmZXe6eFeaMXDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:33.815644Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.06164","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:79998749b649ca845d691552737e19f6f145cc82e1cc73071a02b9cd4ac0ec27","sha256:0b400ad050b3a1ce95f64c03d54b3edfee51502ff4a0b5cfbf8fbb2e0a33acfc"],"state_sha256":"bbfeac1b996128ab429987e0cdb66745c0c7a1df4beee93c91b1abfb31950d7f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DDslQ4c0tVcAkFidIAWqtyULLckulrXzrufZ9vpS+rUhZJS6+gEdpYj4DKtUKDLn/ITRLROJy9swvv9FIZcpDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T15:28:30.416485Z","bundle_sha256":"b2b71312bb741f8b99adce3fe9f0b9627aac0f5fd9a565c139fdf5de3cf4496b"}}