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We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 < $\\alpha$ < $\\beta$ and moreover there is a constant C independent of the initial data such that $\\forall$t > 0, u (t) 2 + |u(t)| $\\beta$+2 $\\le$ C max{t -- 2 $\\alpha$ , t -- ($\\alpha$+1)($\\beta$+2) $\\beta$--$\\alpha$ }."},"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":"1811.02212","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-11-06T08:04:42Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"191541bfc285f640d9a0a2317b0c7b96fd03ed9706c6af8898e58531a722c0a2","abstract_canon_sha256":"4bc49d5d5ebe9a0934503a4936e6f803a3c035d1eed07ce76d0db4ceec134b11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:24.813597Z","signature_b64":"5V4o2Ucs2Q+I/cgyw2EbvzDsBUfsoNsCBTNQfkODtpmuCAcJYIzsvUYPiJa8q+1ZPh+hUkTb3uOGJvYhfCZVAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab83578cfe24a57c5b8ebecb62eeb3bfa893e7738a34a61c10cf0f98468685d5","last_reissued_at":"2026-05-18T00:01:24.813144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:24.813144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The universal bound property for a class of second order ODEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"Alain Haraux (LJLL), Mama Abdelli","submitted_at":"2018-11-06T08:04:42Z","abstract_excerpt":"We consider the scalar second order ODE u + |u | $\\alpha$ u + |u| $\\beta$ u = 0, where $\\alpha$, $\\beta$ are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 < $\\alpha$ < $\\beta$ and moreover there is a constant C independent of the initial data such that $\\forall$t > 0, u (t) 2 + |u(t)| $\\beta$+2 $\\le$ C max{t -- 2 $\\alpha$ , t -- ($\\alpha$+1)($\\beta$+2) $\\beta$--$\\alpha$ }."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02212","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":"1811.02212","created_at":"2026-05-18T00:01:24.813212+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.02212v1","created_at":"2026-05-18T00:01:24.813212+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.02212","created_at":"2026-05-18T00:01:24.813212+00:00"},{"alias_kind":"pith_short_12","alias_value":"VOBVPDH6ESSX","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_16","alias_value":"VOBVPDH6ESSXYW4O","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_8","alias_value":"VOBVPDH6","created_at":"2026-05-18T12:32:59.047623+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/VOBVPDH6ESSXYW4OX3FWF3VTX6","json":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6.json","graph_json":"https://pith.science/api/pith-number/VOBVPDH6ESSXYW4OX3FWF3VTX6/graph.json","events_json":"https://pith.science/api/pith-number/VOBVPDH6ESSXYW4OX3FWF3VTX6/events.json","paper":"https://pith.science/paper/VOBVPDH6"},"agent_actions":{"view_html":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6","download_json":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6.json","view_paper":"https://pith.science/paper/VOBVPDH6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.02212&json=true","fetch_graph":"https://pith.science/api/pith-number/VOBVPDH6ESSXYW4OX3FWF3VTX6/graph.json","fetch_events":"https://pith.science/api/pith-number/VOBVPDH6ESSXYW4OX3FWF3VTX6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6/action/storage_attestation","attest_author":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6/action/author_attestation","sign_citation":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6/action/citation_signature","submit_replication":"https://pith.science/pith/VOBVPDH6ESSXYW4OX3FWF3VTX6/action/replication_record"}},"created_at":"2026-05-18T00:01:24.813212+00:00","updated_at":"2026-05-18T00:01:24.813212+00:00"}