{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:FMMBBFAPG5ATK3ADQU3YPGSSIF","short_pith_number":"pith:FMMBBFAP","schema_version":"1.0","canonical_sha256":"2b1810940f3741356c038537879a52416a275226f5f3873640d8fab1e4629ada","source":{"kind":"arxiv","id":"1809.09531","version":1},"attestation_state":"computed","paper":{"title":"On the energy decay rates for the 1D damped fractional Klein-Gordon equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Milena Stanislavova, Satbir Malhi","submitted_at":"2018-09-25T15:00:31Z","abstract_excerpt":"We consider the fractional Klein-Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non-trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial rate $O(t^{-\\frac{s}{4-2s}})$ for $0< s<2 $ and at some exponential rate when $s\\geq 2$. Our approach is based on the asymptotic theory of $C_0$ semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for th"},"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":"1809.09531","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-25T15:00:31Z","cross_cats_sorted":[],"title_canon_sha256":"e5b638ec234c313e070cd7f08fa4cdea25c0950d12339bfadb073ef53aafac45","abstract_canon_sha256":"130c4c11bcf9fb95deaacd3d0a18777c1e0bed9990c30faac6d0539fd67d5176"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:50.868618Z","signature_b64":"2fy96Zp6ZBqP3lpHZnDRXLhebL77ata8B3Duiw670kJ9t4496QjTegEBTyRX5cyBkB0G/5+tdME4fSnFvmkGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b1810940f3741356c038537879a52416a275226f5f3873640d8fab1e4629ada","last_reissued_at":"2026-05-18T00:04:50.868112Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:50.868112Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the energy decay rates for the 1D damped fractional Klein-Gordon equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Milena Stanislavova, Satbir Malhi","submitted_at":"2018-09-25T15:00:31Z","abstract_excerpt":"We consider the fractional Klein-Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non-trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial rate $O(t^{-\\frac{s}{4-2s}})$ for $0< s<2 $ and at some exponential rate when $s\\geq 2$. Our approach is based on the asymptotic theory of $C_0$ semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.09531","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.09531","created_at":"2026-05-18T00:04:50.868186+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.09531v1","created_at":"2026-05-18T00:04:50.868186+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.09531","created_at":"2026-05-18T00:04:50.868186+00:00"},{"alias_kind":"pith_short_12","alias_value":"FMMBBFAPG5AT","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"FMMBBFAPG5ATK3AD","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"FMMBBFAP","created_at":"2026-05-18T12:32:22.470017+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/FMMBBFAPG5ATK3ADQU3YPGSSIF","json":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF.json","graph_json":"https://pith.science/api/pith-number/FMMBBFAPG5ATK3ADQU3YPGSSIF/graph.json","events_json":"https://pith.science/api/pith-number/FMMBBFAPG5ATK3ADQU3YPGSSIF/events.json","paper":"https://pith.science/paper/FMMBBFAP"},"agent_actions":{"view_html":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF","download_json":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF.json","view_paper":"https://pith.science/paper/FMMBBFAP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.09531&json=true","fetch_graph":"https://pith.science/api/pith-number/FMMBBFAPG5ATK3ADQU3YPGSSIF/graph.json","fetch_events":"https://pith.science/api/pith-number/FMMBBFAPG5ATK3ADQU3YPGSSIF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF/action/storage_attestation","attest_author":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF/action/author_attestation","sign_citation":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF/action/citation_signature","submit_replication":"https://pith.science/pith/FMMBBFAPG5ATK3ADQU3YPGSSIF/action/replication_record"}},"created_at":"2026-05-18T00:04:50.868186+00:00","updated_at":"2026-05-18T00:04:50.868186+00:00"}