{"paper":{"title":"On a non-homogeneous and non-linear heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luca Bisconti, Matteo Franca","submitted_at":"2015-03-09T11:33:03Z","abstract_excerpt":"We consider the Cauchy-problem for a parabolic equation of the following type:\n  \\begin{equation*}\n  \\frac{\\partial u}{\\partial t}= \\Delta u+ f(u,|x|),\n  \\end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial condition $u(x,0)=\\phi$, and we allow $\\phi$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;\\phi)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, Ground States with slow decay lie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02432","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"}