{"paper":{"title":"Gradient estimates and Liouville type theorems for Poisson equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nguyen Ngoc Khanh, Nguyen Thac Dung","submitted_at":"2018-03-20T04:25:23Z","abstract_excerpt":"In this paper, we will address to the following parabolic equation $$ u_t=\\Delta_fu + F(u) $$ on a smooth metric measure space with Bakry-\\'{E}mery curvature bounded from below. Here $F$ is a differentiable function defined in $\\mathbb{R}$. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equations (\\cite{Bai17, CLPW17}). In this paper, we show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either $F=cu(1-u)$ (the Fisher equation) or; $F=-u^3+u$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07251","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"}