Gradient estimates for a general type of nonlinear parabolic equations under geometric conditions and related problems. (January 2023)

• Record Type: Journal Article
• Authors: Dung, Ha Tuan
• Is Part Of: Nonlinear analysis. Volume 226(2023)
• Publication Date: 2023-01
• Subjects: primary 32M05 -- secondary 32H02; Super Perelman–Ricci flows -- Yamabe flows -- Smooth metric measure spaces -- Lichnerowicz type equations -- Yamabe type equations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113135
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we establish gradient estimates for the positive bounded solutions to a general type of nonlinear parabolic equation concerning the weighted Laplacian ∂ ∂ t − a ( x, t ) − Δ f u ( x, t ) = F ( u ( x, t ) ) on a smooth metric measure space with the metric evolving under the ( k, ∞ ) -super Perelman–Ricci flow and the Yamabe flow. Applications of our results include Liouville type results and gradient estimates for some important geometric partial differential equations such as the equations involving gradient Ricci solitons and the Einstein-scalar field Lichnerowicz type equations. Our results generalize and improve many previous works.