{"paper":{"title":"Extinction profile of the logarithmic diffusion equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui, SungHoon Kim","submitted_at":"2010-12-09T03:23:55Z","abstract_excerpt":"Let $u$ be the solution of $u_t=\\Delta\\log u$ in $\\R^N\\times (0,T)$, N=3 or $N\\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\\le u_0\\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\\log (T-t)$, converges uniformly on $\\R^N$ to the rescaled Barenblatt solution $\\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\\to\\infty$. We also obtain convergence of the rescaled solution "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1915","kind":"arxiv","version":4},"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"}