{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:462CAXZPMHCRMWFP4G73XFUL6Q","short_pith_number":"pith:462CAXZP","canonical_record":{"source":{"id":"1709.05785","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-18T06:23:34Z","cross_cats_sorted":[],"title_canon_sha256":"8c1789a68583400f3588eddd8780b76ca9b956fb895af700a6787a30f99411ec","abstract_canon_sha256":"ed855760f8ad536b3b3cede570e735a881d002b7cc10c9b11ebd2f977941ce62"},"schema_version":"1.0"},"canonical_sha256":"e7b4205f2f61c51658afe1bfbb968bf418f11413303c6b82b1b025dffa3baee5","source":{"kind":"arxiv","id":"1709.05785","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05785","created_at":"2026-05-18T00:19:03Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05785v3","created_at":"2026-05-18T00:19:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05785","created_at":"2026-05-18T00:19:03Z"},{"alias_kind":"pith_short_12","alias_value":"462CAXZPMHCR","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"462CAXZPMHCRMWFP","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"462CAXZP","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:462CAXZPMHCRMWFP4G73XFUL6Q","target":"record","payload":{"canonical_record":{"source":{"id":"1709.05785","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-18T06:23:34Z","cross_cats_sorted":[],"title_canon_sha256":"8c1789a68583400f3588eddd8780b76ca9b956fb895af700a6787a30f99411ec","abstract_canon_sha256":"ed855760f8ad536b3b3cede570e735a881d002b7cc10c9b11ebd2f977941ce62"},"schema_version":"1.0"},"canonical_sha256":"e7b4205f2f61c51658afe1bfbb968bf418f11413303c6b82b1b025dffa3baee5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:03.561506Z","signature_b64":"ehEum1Skeoik3j4q/8XZoh+OqiY7ZUBxE4h4CX7LpNVjTkmxo+C8lYUmp9NSXVtQXIeTz2RQ6vF+HEQ4MNhXDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7b4205f2f61c51658afe1bfbb968bf418f11413303c6b82b1b025dffa3baee5","last_reissued_at":"2026-05-18T00:19:03.560803Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:03.560803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.05785","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:19:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"da3jQ1wRj5GwbOgM8LyTaJyujeQ/iVg7OC39dS+meVO3WtnFgoc618jaGQooO0dW1ysxnJPfup56l4wse2OtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T23:25:55.393450Z"},"content_sha256":"924a18782f9fe63efd10ce77fb055a6b37c82153ec0ea3fbb33352f675cc170f","schema_version":"1.0","event_id":"sha256:924a18782f9fe63efd10ce77fb055a6b37c82153ec0ea3fbb33352f675cc170f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:462CAXZPMHCRMWFP4G73XFUL6Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\\mathbb{R}^N$. I. Persistence and asymptotic spreading","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rachidi B. Salako, Wenxian Shen","submitted_at":"2017-09-18T06:23:34Z","abstract_excerpt":"The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$\\partial_tu=\\Delta u-\\chi\\nabla(u\\nabla v)+u(a(x,t)-ub(x,t))\\ ,\\ 0=\\Delta v-\\lambda v+\\mu u \\ \\ (1)$$where $\\chi, \\lambda, \\mu$ are positive constants, $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading. Under some explicit condition on the parameters, we show that (1) has a unique nonnegative time global classical solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every $t_0\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05785","kind":"arxiv","version":3},"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":"2026-05-18T00:19:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vksmpwrg41lHLBNDdf0Xs30eJvsG4IsWGiB6BEY0SzHDKg3n2iD2wLVlIg5qfckPIvWos4XIc2T4QEhy4Q7aAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T23:25:55.394026Z"},"content_sha256":"5debdbb9d79950cd1982a95e748199ffb2be15d9943b805302962059fef65cb2","schema_version":"1.0","event_id":"sha256:5debdbb9d79950cd1982a95e748199ffb2be15d9943b805302962059fef65cb2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/462CAXZPMHCRMWFP4G73XFUL6Q/bundle.json","state_url":"https://pith.science/pith/462CAXZPMHCRMWFP4G73XFUL6Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/462CAXZPMHCRMWFP4G73XFUL6Q/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":"2026-05-31T23:25:55Z","links":{"resolver":"https://pith.science/pith/462CAXZPMHCRMWFP4G73XFUL6Q","bundle":"https://pith.science/pith/462CAXZPMHCRMWFP4G73XFUL6Q/bundle.json","state":"https://pith.science/pith/462CAXZPMHCRMWFP4G73XFUL6Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/462CAXZPMHCRMWFP4G73XFUL6Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:462CAXZPMHCRMWFP4G73XFUL6Q","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":"ed855760f8ad536b3b3cede570e735a881d002b7cc10c9b11ebd2f977941ce62","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-18T06:23:34Z","title_canon_sha256":"8c1789a68583400f3588eddd8780b76ca9b956fb895af700a6787a30f99411ec"},"schema_version":"1.0","source":{"id":"1709.05785","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.05785","created_at":"2026-05-18T00:19:03Z"},{"alias_kind":"arxiv_version","alias_value":"1709.05785v3","created_at":"2026-05-18T00:19:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05785","created_at":"2026-05-18T00:19:03Z"},{"alias_kind":"pith_short_12","alias_value":"462CAXZPMHCR","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"462CAXZPMHCRMWFP","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"462CAXZP","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:5debdbb9d79950cd1982a95e748199ffb2be15d9943b805302962059fef65cb2","target":"graph","created_at":"2026-05-18T00:19:03Z","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":"The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$\\partial_tu=\\Delta u-\\chi\\nabla(u\\nabla v)+u(a(x,t)-ub(x,t))\\ ,\\ 0=\\Delta v-\\lambda v+\\mu u \\ \\ (1)$$where $\\chi, \\lambda, \\mu$ are positive constants, $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading. Under some explicit condition on the parameters, we show that (1) has a unique nonnegative time global classical solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every $t_0\\","authors_text":"Rachidi B. Salako, Wenxian Shen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-18T06:23:34Z","title":"Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\\mathbb{R}^N$. I. Persistence and asymptotic spreading"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05785","kind":"arxiv","version":3},"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":"sha256:924a18782f9fe63efd10ce77fb055a6b37c82153ec0ea3fbb33352f675cc170f","target":"record","created_at":"2026-05-18T00:19:03Z","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":"ed855760f8ad536b3b3cede570e735a881d002b7cc10c9b11ebd2f977941ce62","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-18T06:23:34Z","title_canon_sha256":"8c1789a68583400f3588eddd8780b76ca9b956fb895af700a6787a30f99411ec"},"schema_version":"1.0","source":{"id":"1709.05785","kind":"arxiv","version":3}},"canonical_sha256":"e7b4205f2f61c51658afe1bfbb968bf418f11413303c6b82b1b025dffa3baee5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e7b4205f2f61c51658afe1bfbb968bf418f11413303c6b82b1b025dffa3baee5","first_computed_at":"2026-05-18T00:19:03.560803Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:03.560803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ehEum1Skeoik3j4q/8XZoh+OqiY7ZUBxE4h4CX7LpNVjTkmxo+C8lYUmp9NSXVtQXIeTz2RQ6vF+HEQ4MNhXDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:03.561506Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.05785","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:924a18782f9fe63efd10ce77fb055a6b37c82153ec0ea3fbb33352f675cc170f","sha256:5debdbb9d79950cd1982a95e748199ffb2be15d9943b805302962059fef65cb2"],"state_sha256":"2b2010c9f8d4c8eddcc69089fb0bb54f254ffca5008bde0b31f02236e33b91e1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6j5mQKIPgAvLw/JHZeffescXahZMYb8YyIZObIrB/vyw0H3kJRLkGfRTX1KXvUYv2p/jooFNEzGSMn5+O/x2Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T23:25:55.397648Z","bundle_sha256":"50932405f3753f15cbe07054bb52e9b2f3c4bb3d0598b01141aeec54e4a061ce"}}