{"paper":{"title":"Improvements on lower bounds for the blow-up time under local nonlinear Neumann conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xin Yang, Zhengfang Zhou","submitted_at":"2017-07-06T05:20:02Z","abstract_excerpt":"This paper studies the heat equation $u_t=\\Delta u$ in a bounded domain $\\Omega\\subset\\mathbb{R}^{n}(n\\geq 2)$ with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative $\\partial u/\\partial n=u^{q}$ on partial boundary $\\Gamma_1\\subseteq \\partial\\Omega$ for some $q>1$, while $\\partial u/\\partial n=0$ on the other part. We investigate the lower bound of the blow-up time $T^{*}$ of $u$ in several aspects. First, $T^{*}$ is proved to be at least of order $(q-1)^{-1}$ as $q\\rightarrow 1^{+}$. Since the existing upper bound is of order $(q-1)^{-1}$, this res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01641","kind":"arxiv","version":2},"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"}