{"paper":{"title":"Positivity, decay, and extinction for a singular diffusion equation with gradient absorption","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"IMAR), Philippe Laurencot (IMT), Razvan Gabriel Iagar (IMT","submitted_at":"2011-04-08T08:03:16Z","abstract_excerpt":"We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \\partial_t u -\\Delta_{p}u+|\\nabla u|^{q}=0\\quad in\\;\\; (0,\\infty)\\times\\RR^N, where $N\\ge 1$, $p\\in(1,2)$, and $q>0$. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as $t\\to\\infty$ for $q>p-N/(N+1)$, optimal decay estimates as $t\\to\\infty$ for $p/2\\le q\\le p-N/(N+1)$, or extinction in finite time for $0 < q < p/2$. In addition, we show how the diffusion prevents extinction "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1513","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"}