{"paper":{"title":"Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tobias Black","submitted_at":"2015-10-24T18:32:08Z","abstract_excerpt":"We study nonnnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel whole space system \\begin{align*} \\left\\{\\begin{array}{c@{\\,}l@{\\quad}l@{\\,}c} u_{t}&=\\Delta u-\\nabla\\!\\cdot(u\\nabla v),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ 0 &=\\Delta v+u+f(x),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ u(x,0)&=u_{0}(x),\\ &x\\in\\mathbb{R}^n,&\n  \\end{array}\\right. \\end{align*} with prototypical external signal production \\begin{align*} f(x):=\\begin{cases} f_0\\vert x\\vert^{-\\alpha},&\\text{ if }\\vert x\\vert \\leq R-\\rho,\\\\ 0,&\\text{ if } \\vert x\\vert\\geq R+\\rho,\\\\ \\end{cases} \\end{align*} for $R\\in(0,1)$ and $\\rho\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07173","kind":"arxiv","version":1},"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"}