{"paper":{"title":"Well-posedness, Global existence and decay estimates for the heat equation with general power-exponential nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mohamed Majdoub, Slim Tayachi","submitted_at":"2016-07-10T09:36:20Z","abstract_excerpt":"In this paper we consider the problem: $\\partial_{t} u- \\Delta u=f(u),\\; u(0)=u_0\\in \\exp L^p(\\R^N),$ where $p>1$ and $f : \\R\\to\\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\\exp L^p_0(\\R^N)$ for $f(u)\\sim \\mbox{e}^{|u|^q},\\;0<q\\leq p,\\; |u|\\to \\infty.$ However, if for some $\\lambda>0,$ $\\displaystyle\\liminf_{s\\to \\infty}\\left(f(s)\\,{\\rm{e}}^{-\\lambda s^p}\\right)>0,$ then non-existence occurs in $\\exp L^p(\\R^N).$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|\\sim |u|^{m}$ as $u\\to 0,$ ${N(m-1)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02723","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"}