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This model was recently proposed by Fujie et al. \\cite{FIY14} as a model for tumor invasion with the role of extracellular matrix incorporated, and was analyzed by Fujie et al. \\cite{FIW"},"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":"1604.03898","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-13T18:13:23Z","cross_cats_sorted":[],"title_canon_sha256":"24702b76f5a86e29a5472fcbbd35b3aeaad39fba32a544a8803ce39390f94613","abstract_canon_sha256":"dc988a867a25dd338dada45fd5d31d7d71077e92a300ca233fb2f0d31945f6cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:41.735306Z","signature_b64":"ZsACQ1SUvU2tnYGVaUQxtrWP6pw53XFFjc8jyD1F9acXFY2lx0nVbU90xCS3aGg6uzHFcRE8/u+Y0dlqtgEFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fc4463ccdf4dc89b4b5ec0aed5dfb656d0c9f14dc6e171a4021b5ef49fd39b6","last_reissued_at":"2026-05-18T00:10:41.734723Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:41.734723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundedness and exponential convergence of a chemotaxis model for tumor invasion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Haiyang Jin, Tian Xiang","submitted_at":"2016-04-13T18:13:23Z","abstract_excerpt":"We revisit the following chemotaxis system modeling tumor invasion \\begin{equation*} \\begin{cases} u_t=\\Delta u-\\nabla \\cdot(u\\nabla v),& x\\in\\Omega, t>0,\\\\ v_t=\\Delta v+wz,& x\\in\\Omega, t>0,\\\\ w_t=-wz,& x\\in\\Omega, t>0,\\\\ z_t=\\Delta z-z+u, & x\\in\\Omega, t>0,\\\\ \\end{cases} \\end{equation*} in a smooth bounded domain $\\Omega \\subset \\mathbb{R}^n(n\\geq 1)$ with homogeneous Neumann boundary and initial conditions. 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