{"paper":{"title":"Gradient estimates for some $f$-heat equations driven by Lichnerowicz's equation on complete smooth metric measure spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Nguyen Ngoc Khanh, Nguyen Thac Dung, Qu\\^oc Anh Ng\\^o","submitted_at":"2016-10-11T05:51:28Z","abstract_excerpt":"Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-\\'Emery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $$ u_t=\\Delta_f u+au\\log u+bu +Au^p+Bu^{-q} $$ and \\[ u_t=\\Delta_f u+Ae^{pu}+Be^{-pu}+D \\] are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schr\\\"{o}dinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03199","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"}