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Salako, Shuwen Xue, Wenxian Shen","submitted_at":"2018-12-28T06:07:15Z","abstract_excerpt":"The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, \\begin{equation}\\label{abstract-eq1} \\begin{cases} u_t=u_{xx}-\\chi(uv_x)_x +u(a-bu),\\quad x\\in\\R\\cr 0=v_{xx}- \\lambda v+\\mu u,\\quad x\\in\\R, \\end{cases} \\end{equation} where $\\chi$, $a$, $b$, $\\lambda$, and $\\mu$ are positive constants.\n  Assume $b>\\chi\\mu$. Then if in addition, $\\big(1+\\frac{1}{2}\\frac{(\\sqrt{a}-\\sqrt{\\lambda})_+}{(\\sqrt{a}+\\sqrt{\\la})}\\big)\\chi\\mu { \\leq} b$ holds, it is proved that $c_0^*=2\\sqrt a$ is the spreading speed of the solution"},"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":"1901.00045","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-28T06:07:15Z","cross_cats_sorted":[],"title_canon_sha256":"ed5cb3c66485d4647befdb29bd07bbc81cc2a73fa78de7e77edd01b1656f2c9e","abstract_canon_sha256":"61b724d2cd13ed8fc407259de5ee42e4cf808623540569656f417df8f82b8239"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:15.724877Z","signature_b64":"hmfpFwRFcZft1uZAp7/E19Sf1hRWYkQBzqw0aYR9UQev6s+CtZX+s3ilKsrcBDc7KoVzj0FEQ8u5hG4241OBAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e62d2b1d251f90b2a83e470498c6e732bf744ed9622ed05c0d2d67bdb141130c","last_reissued_at":"2026-05-17T23:47:15.724480Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:15.724480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rachidi B. Salako, Shuwen Xue, Wenxian Shen","submitted_at":"2018-12-28T06:07:15Z","abstract_excerpt":"The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, \\begin{equation}\\label{abstract-eq1} \\begin{cases} u_t=u_{xx}-\\chi(uv_x)_x +u(a-bu),\\quad x\\in\\R\\cr 0=v_{xx}- \\lambda v+\\mu u,\\quad x\\in\\R, \\end{cases} \\end{equation} where $\\chi$, $a$, $b$, $\\lambda$, and $\\mu$ are positive constants.\n  Assume $b>\\chi\\mu$. Then if in addition, $\\big(1+\\frac{1}{2}\\frac{(\\sqrt{a}-\\sqrt{\\lambda})_+}{(\\sqrt{a}+\\sqrt{\\la})}\\big)\\chi\\mu { \\leq} b$ holds, it is proved that $c_0^*=2\\sqrt a$ is the spreading speed of the solution"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.00045","kind":"arxiv","version":2},"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":"1901.00045","created_at":"2026-05-17T23:47:15.724550+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.00045v2","created_at":"2026-05-17T23:47:15.724550+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.00045","created_at":"2026-05-17T23:47:15.724550+00:00"},{"alias_kind":"pith_short_12","alias_value":"4YWSWHJFD6IL","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"4YWSWHJFD6ILFKB6","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"4YWSWHJF","created_at":"2026-05-18T12:32:05.422762+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/4YWSWHJFD6ILFKB6I4CJRRXHGK","json":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK.json","graph_json":"https://pith.science/api/pith-number/4YWSWHJFD6ILFKB6I4CJRRXHGK/graph.json","events_json":"https://pith.science/api/pith-number/4YWSWHJFD6ILFKB6I4CJRRXHGK/events.json","paper":"https://pith.science/paper/4YWSWHJF"},"agent_actions":{"view_html":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK","download_json":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK.json","view_paper":"https://pith.science/paper/4YWSWHJF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.00045&json=true","fetch_graph":"https://pith.science/api/pith-number/4YWSWHJFD6ILFKB6I4CJRRXHGK/graph.json","fetch_events":"https://pith.science/api/pith-number/4YWSWHJFD6ILFKB6I4CJRRXHGK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK/action/storage_attestation","attest_author":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK/action/author_attestation","sign_citation":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK/action/citation_signature","submit_replication":"https://pith.science/pith/4YWSWHJFD6ILFKB6I4CJRRXHGK/action/replication_record"}},"created_at":"2026-05-17T23:47:15.724550+00:00","updated_at":"2026-05-17T23:47:15.724550+00:00"}