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Salako, Wenxian Shen","submitted_at":"2016-12-03T05:01:01Z","abstract_excerpt":"We consider the following chemotaxis systems $$\\begin{cases}u_t=\\Delta u-\\chi_1\\nabla(u\\nabla v_1)+\\chi_2\\nabla(u\\nabla v_2)+u(a-bu),\\ \\ x\\in\\mathbb R^N,t>0,\\\\0=(\\Delta-\\lambda_1I)v_1+\\mu_1u,\\ \\ x\\in\\mathbb R^N,t>0,\\\\0=(\\Delta-\\lambda_2I)v_2+\\mu_2u,\\ \\ \\text{in}\\ x\\in\\mathbb R^N,\\ t>0,\\\\u(\\cdot,0)=u_0,\\ \\ x\\in\\mathbb R^N,\\end{cases}$$where $\\chi_i,\\ \\lambda_i,\\ \\mu_i,\\ i=1,2$ and $a,\\ b$ are positive constant real numbers and $N$ is a positive integer. Under some conditions on the parameters, we prove the global existence and boundedness of classical solutions $(u(x,t;u_0),v_1(x,t;u_0),v_2(x,t"},"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":"1612.00924","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-03T05:01:01Z","cross_cats_sorted":[],"title_canon_sha256":"f052c6880a3d88f074ecbe519d010d222140dc5a0353a3fd3dc190503170abe2","abstract_canon_sha256":"4e60d277705785d570d410978fe42e233675f1a6e1bda78cef0676b791072b24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:53.316602Z","signature_b64":"bmUkM6sbwcSQboaFrbALDrSOeRraczh/XMr2mgfsMLodDJDXAaWXky05m1xcQocz1NwST55GIT9Dy/q+Cpa0Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5398ffc0f05436602205d2f673650abca4ea0a9ba2ab3f5e00f692a5600804bd","last_reissued_at":"2026-05-18T00:41:53.316103Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:53.316103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\\mathbb{R}^{N}$","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":"2016-12-03T05:01:01Z","abstract_excerpt":"We consider the following chemotaxis systems $$\\begin{cases}u_t=\\Delta u-\\chi_1\\nabla(u\\nabla v_1)+\\chi_2\\nabla(u\\nabla v_2)+u(a-bu),\\ \\ x\\in\\mathbb R^N,t>0,\\\\0=(\\Delta-\\lambda_1I)v_1+\\mu_1u,\\ \\ x\\in\\mathbb R^N,t>0,\\\\0=(\\Delta-\\lambda_2I)v_2+\\mu_2u,\\ \\ \\text{in}\\ x\\in\\mathbb R^N,\\ t>0,\\\\u(\\cdot,0)=u_0,\\ \\ x\\in\\mathbb R^N,\\end{cases}$$where $\\chi_i,\\ \\lambda_i,\\ \\mu_i,\\ i=1,2$ and $a,\\ b$ are positive constant real numbers and $N$ is a positive integer. Under some conditions on the parameters, we prove the global existence and boundedness of classical solutions $(u(x,t;u_0),v_1(x,t;u_0),v_2(x,t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00924","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.00924","created_at":"2026-05-18T00:41:53.316181+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.00924v3","created_at":"2026-05-18T00:41:53.316181+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.00924","created_at":"2026-05-18T00:41:53.316181+00:00"},{"alias_kind":"pith_short_12","alias_value":"KOMP7QHQKQ3G","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"KOMP7QHQKQ3GAIQF","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"KOMP7QHQ","created_at":"2026-05-18T12:30:25.849896+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/KOMP7QHQKQ3GAIQF2L3HGZIKXS","json":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS.json","graph_json":"https://pith.science/api/pith-number/KOMP7QHQKQ3GAIQF2L3HGZIKXS/graph.json","events_json":"https://pith.science/api/pith-number/KOMP7QHQKQ3GAIQF2L3HGZIKXS/events.json","paper":"https://pith.science/paper/KOMP7QHQ"},"agent_actions":{"view_html":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS","download_json":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS.json","view_paper":"https://pith.science/paper/KOMP7QHQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.00924&json=true","fetch_graph":"https://pith.science/api/pith-number/KOMP7QHQKQ3GAIQF2L3HGZIKXS/graph.json","fetch_events":"https://pith.science/api/pith-number/KOMP7QHQKQ3GAIQF2L3HGZIKXS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS/action/storage_attestation","attest_author":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS/action/author_attestation","sign_citation":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS/action/citation_signature","submit_replication":"https://pith.science/pith/KOMP7QHQKQ3GAIQF2L3HGZIKXS/action/replication_record"}},"created_at":"2026-05-18T00:41:53.316181+00:00","updated_at":"2026-05-18T00:41:53.316181+00:00"}