{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:GMJFDD6IKFYRQ7JGIPEKVH2F42","short_pith_number":"pith:GMJFDD6I","canonical_record":{"source":{"id":"1009.6124","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-30T13:28:07Z","cross_cats_sorted":[],"title_canon_sha256":"007b5a447cc3c16933585bd3118ac412725aee528fb98a65d146e7429cdb8311","abstract_canon_sha256":"ada167a58e0a0a04fe773e779dce951700f970eaad561e57d23a510492e4e371"},"schema_version":"1.0"},"canonical_sha256":"3312518fc85171187d2643c8aa9f45e68fbe85be3348b83e8dab6cc35e09e8f0","source":{"kind":"arxiv","id":"1009.6124","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.6124","created_at":"2026-05-18T04:40:01Z"},{"alias_kind":"arxiv_version","alias_value":"1009.6124v1","created_at":"2026-05-18T04:40:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.6124","created_at":"2026-05-18T04:40:01Z"},{"alias_kind":"pith_short_12","alias_value":"GMJFDD6IKFYR","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GMJFDD6IKFYRQ7JG","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GMJFDD6I","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:GMJFDD6IKFYRQ7JGIPEKVH2F42","target":"record","payload":{"canonical_record":{"source":{"id":"1009.6124","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-30T13:28:07Z","cross_cats_sorted":[],"title_canon_sha256":"007b5a447cc3c16933585bd3118ac412725aee528fb98a65d146e7429cdb8311","abstract_canon_sha256":"ada167a58e0a0a04fe773e779dce951700f970eaad561e57d23a510492e4e371"},"schema_version":"1.0"},"canonical_sha256":"3312518fc85171187d2643c8aa9f45e68fbe85be3348b83e8dab6cc35e09e8f0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:01.025213Z","signature_b64":"SodqTaJbNnzEz1WvMBYTsl1uvawTFt78Tq4PUntHG6oZ5hhygT+VSYKfid/REZ6cGCu6CsN+sHP+5CfFiQHpCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3312518fc85171187d2643c8aa9f45e68fbe85be3348b83e8dab6cc35e09e8f0","last_reissued_at":"2026-05-18T04:40:01.024799Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:01.024799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.6124","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":"2026-05-18T04:40:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Zrc4eg4HRHljMLYO9uhUeoGIgdS/BrQ7slY9Dxchybfe9bqgXJl5VyzCrFa54bwHfyIGxA2NKp+IRT9lElWDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:55:44.548713Z"},"content_sha256":"801f415037b196f191a374dcf59c896f69fc26c090925ae627c22f087abe7fea","schema_version":"1.0","event_id":"sha256:801f415037b196f191a374dcf59c896f69fc26c090925ae627c22f087abe7fea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:GMJFDD6IKFYRQ7JGIPEKVH2F42","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic stability of solutions to abstract differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"A.G.Ramm","submitted_at":"2010-09-30T13:28:07Z","abstract_excerpt":"An evolution problem for abstract differential equations is studied. The typical problem is: $$\\dot{u}=A(t)u+F(t,u), \\quad t\\geq 0; \\,\\, u(0)=u_0;\\quad \\dot{u}=\\frac {du}{dt}\\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\\|F(t,u)\\|\\leq c_0\\|u\\|^p,\\,\\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\\leq -\\gamma(t)\\|u\\|^2$ $\\forall u\\in H$, where $\\gamma(t)>0$, and the case when $\\lim_{t\\to \\infty}\\gamma(t)=0$ is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this es"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.6124","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":"2026-05-18T04:40:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nm3RrIcTOd+m1NsRz1IQEVxQrXODmWQBgJRhozY0Is3qvUa0h2rXWGjmlsAogTHV+ezBjd+7rIAVId59sn8rAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:55:44.549446Z"},"content_sha256":"bb78dd8e0e354d9896714d9b533654585f94859913a7a5d0f1bcf43892e9b893","schema_version":"1.0","event_id":"sha256:bb78dd8e0e354d9896714d9b533654585f94859913a7a5d0f1bcf43892e9b893"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GMJFDD6IKFYRQ7JGIPEKVH2F42/bundle.json","state_url":"https://pith.science/pith/GMJFDD6IKFYRQ7JGIPEKVH2F42/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GMJFDD6IKFYRQ7JGIPEKVH2F42/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":"2026-05-26T02:55:44Z","links":{"resolver":"https://pith.science/pith/GMJFDD6IKFYRQ7JGIPEKVH2F42","bundle":"https://pith.science/pith/GMJFDD6IKFYRQ7JGIPEKVH2F42/bundle.json","state":"https://pith.science/pith/GMJFDD6IKFYRQ7JGIPEKVH2F42/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GMJFDD6IKFYRQ7JGIPEKVH2F42/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:GMJFDD6IKFYRQ7JGIPEKVH2F42","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":"ada167a58e0a0a04fe773e779dce951700f970eaad561e57d23a510492e4e371","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-30T13:28:07Z","title_canon_sha256":"007b5a447cc3c16933585bd3118ac412725aee528fb98a65d146e7429cdb8311"},"schema_version":"1.0","source":{"id":"1009.6124","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.6124","created_at":"2026-05-18T04:40:01Z"},{"alias_kind":"arxiv_version","alias_value":"1009.6124v1","created_at":"2026-05-18T04:40:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.6124","created_at":"2026-05-18T04:40:01Z"},{"alias_kind":"pith_short_12","alias_value":"GMJFDD6IKFYR","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GMJFDD6IKFYRQ7JG","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GMJFDD6I","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:bb78dd8e0e354d9896714d9b533654585f94859913a7a5d0f1bcf43892e9b893","target":"graph","created_at":"2026-05-18T04:40:01Z","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":"An evolution problem for abstract differential equations is studied. The typical problem is: $$\\dot{u}=A(t)u+F(t,u), \\quad t\\geq 0; \\,\\, u(0)=u_0;\\quad \\dot{u}=\\frac {du}{dt}\\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\\|F(t,u)\\|\\leq c_0\\|u\\|^p,\\,\\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\\leq -\\gamma(t)\\|u\\|^2$ $\\forall u\\in H$, where $\\gamma(t)>0$, and the case when $\\lim_{t\\to \\infty}\\gamma(t)=0$ is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this es","authors_text":"A.G.Ramm","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-30T13:28:07Z","title":"Asymptotic stability of solutions to abstract differential equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.6124","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":"sha256:801f415037b196f191a374dcf59c896f69fc26c090925ae627c22f087abe7fea","target":"record","created_at":"2026-05-18T04:40:01Z","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":"ada167a58e0a0a04fe773e779dce951700f970eaad561e57d23a510492e4e371","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-30T13:28:07Z","title_canon_sha256":"007b5a447cc3c16933585bd3118ac412725aee528fb98a65d146e7429cdb8311"},"schema_version":"1.0","source":{"id":"1009.6124","kind":"arxiv","version":1}},"canonical_sha256":"3312518fc85171187d2643c8aa9f45e68fbe85be3348b83e8dab6cc35e09e8f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3312518fc85171187d2643c8aa9f45e68fbe85be3348b83e8dab6cc35e09e8f0","first_computed_at":"2026-05-18T04:40:01.024799Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:01.024799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SodqTaJbNnzEz1WvMBYTsl1uvawTFt78Tq4PUntHG6oZ5hhygT+VSYKfid/REZ6cGCu6CsN+sHP+5CfFiQHpCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:01.025213Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.6124","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:801f415037b196f191a374dcf59c896f69fc26c090925ae627c22f087abe7fea","sha256:bb78dd8e0e354d9896714d9b533654585f94859913a7a5d0f1bcf43892e9b893"],"state_sha256":"d922854236bc711205eb8a90bd91379e61aa2770876cee65c0ce0c426b326d91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9IU2YbcIjT/o1D0vkalmuGbkPYs0PMijow7bUGfQ1DAUTt2FcRXCocssxS9WoKzKeAi64yFY7T78Bvli6AUvDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T02:55:44.553107Z","bundle_sha256":"873758cccc219a04d9757049e87cf3d6e5bb4a68cfc35d67922aeb8c0948ce58"}}