{"paper":{"title":"Modica type gradient estimates for reaction-diffusion equations and a parabolic counterpart of a conjecture of De Giorgi","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Agnid Banerjee, Nicola Garofalo","submitted_at":"2015-02-20T01:01:29Z","abstract_excerpt":"We continue the study of Modica type gradient estimates for non-homogeneous parabolic equations initiated in \\cite{BG}. First, we show that for the parabolic minimal surface equation with a semilinear force term if a certain gradient estimate is satisfied at $t=0$, then it holds for all later times $t>0$. We then establish analogous results for reaction-diffusion equations such as \\eqref{e0} below in $\\Om \\times [0, T]$, where $\\Om$ is an epigraph such that the mean curvature of $\\partial \\Om$ is nonnegative.\n  We then turn our attention to settings where such gradient estimates are valid with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.05758","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"}