{"paper":{"title":"Long time existence of Smooth solution for the porous medium equation in a bounded domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"SungHoon Kim","submitted_at":"2012-05-30T15:06:29Z","abstract_excerpt":"In this paper, we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain: {equation} {cases} u_t=\\La u^m\\quad\\text{in $\\Omega\\times [0,\\infty)$} u(x,0)=u_0>0\\quad\\text{in $\\Omega$} u(x,t)=0\\quad\\text{for $x\\in\\partial\\Omega$} {cases} {equation} where $m>1$ is the permeability. The proof is based on the short time existence of $C^{2,\\bar{\\gamma}}_{s}$-smooth solution, the global $C^{1}_{s}$-estimate, the H\\\"older estimate of divergence type degenerate equation with measurable coefficients and $C^{1,\\bar{\\gamma}}_s$-estimate of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.6716","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"}