Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces
The purpose of the article under review is the investigation of gradient estimates under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities, as well as general Liouville-type and other global constancy results. Then the authors proved gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces.\N\NSeveral applications and consequences are presented and discussed.\N\NMore precisely, in Section 2 background material on smooth metric measure spaces and the associated generalised Ricci curvature tensors as required for the development of the paper is presented. Section 3 contains the main results of the paper along with some related discussion. In Section 4 the proof of the local Souplet-Zhang type gradient estimate is given in Theorem 3.1 and in Section 5 the authors give the proof of the local and global elliptic Harnack inequality in Theorem 3.3. Section 6 presents the proof of the local Hamilton-type gradient estimate in Theorem 3.4 and in Section 7 the proof of the various Liouville-type results (in Theorems 3.7 and 3.8.) is given. In Section 8 the authors gave the local Li-Yau differential Harnack estimate in Theorem 3.12 and in Section 9 they establish the local and global parabolic Harnack inequalities (Theorem 3.14.). Section 10 is devoted to the proof of the general Liouville result in Theorem 3.16 and its Corollaries.\N\NThe article is interesting and well-written, with detailed proofs, clearly presented. The paper ends with a comprehensive references list.
scientific article; zbMATH DE number 7857774
scientific article