{"paper":{"title":"Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Winkler, Youshan Tao","submitted_at":"2011-06-27T10:11:57Z","abstract_excerpt":"We consider the quasilinear parabolic-parabolic Keller-Segel system $$\n u_t=\\nabla \\cdot (D(u)\\nabla u) - \\nabla \\cdot (S(u)\\nabla v),\n  \\qquad x\\in\\Omega, \\ t>0,\n v_t=\\Delta v -v + u,\n  x\\in\\Omega, \\ t>0,\n$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\\Omega\\subset\\R^n$ with $n\\ge 2$.\nIt is proved that if $\\frac{S(u)}{D(u)}\\le cu^{\\alpha}$ with $\\alpha<\\frac{2}{n}$ and some constant $c>0$ for all $u>1$ and some further technical conditions are fulfilled, then the classical solutions to the above system are uniformly-in-time bounded. This boundedness result is opt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5345","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"}