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- Differential Harnack inequalities for Fisher-KPP type equations on Riemannian manifolds(arXiv)
Author : Zhihao Lu
Abstract : We obtain almost optimal differential Harnack inequalities for a class of nonlinear parabolic equations on Riemannian manifolds with Bakry-Émery Ricci curvature bounded below, which includes the classical Fisher-KPP equation and Newell-Whitehead equation. Compared to existing research, we do not impose any additional conditions on the positive solutions. As its application, we derive some optimal Liouville properties.
2. Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces(arXiv)
Author : : Ali Taheri, Vahideh Vahidifar
Abstract : In this paper we prove 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. These estimates are established 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 elliptic and parabolic Liouville-type and other global constancy results. Several applications and consequences are presented and discussed