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It is shown that under appropriate regularity assumption on the initial data $(u_0, v_0, w_0)$, the corr"},"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":"1407.7382","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-07-28T11:20:06Z","cross_cats_sorted":[],"title_canon_sha256":"1a26616360a81f4675ec653b36c9f0f4e8eb600e2fad21fa2ca1558767b6f4ba","abstract_canon_sha256":"8573d1a4f8efe29d84e5ed1c9bd4065d4ff0c3eec8c96d1fa1658ca38672b7f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:23.249929Z","signature_b64":"TMsdrIKJqJXHJQ3UIRt7o2P9wXWesk2OFyXChe0gRj/TOL7VcDfsRvk+feItN1R5J2/gGQ/nqyjizNX0HB0xDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a42d20ec88f92f23fdbca8408c4829c88b45cf3ff4c977ebaf62fcf23178929","last_reissued_at":"2026-05-18T02:46:23.249536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:23.249536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundedness in a two-dimensional chemotaxis-haptotaxis system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Youshan Tao","submitted_at":"2014-07-28T11:20:06Z","abstract_excerpt":"This work studies the chemotaxis-haptotaxis system\n  $$\\left\\{ \\begin{array}{ll} u_t= \\Delta u - \\chi \\nabla \\cdot (u\\nabla v) - \\xi \\nabla \\cdot (u\\nabla w) + \\mu u(1-u-w), &\\qquad x\\in \\Omega, \\, t>0, \\\\[1mm]\n  v_t=\\Delta v-v+u, &\\qquad x\\in \\Omega, \\, t>0, \\\\[1mm]\n  w_t=-vw, &\\qquad x\\in \\Omega, \\, t>0, \\end{array} \\right. $$\n   in a bounded smooth domain $\\Omega\\subset\\mathbb{R}^2$ with zero-flux boundary conditions, where the parameters $\\chi, \\xi$ and $\\mu$ are assumed to be positive. 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