Upper bounds of Hessian matrix and gradient estimates of positive solutions to the nonlinear parabolic equation along Ricci flow. (January 2022)

• Record Type: Journal Article
• Authors: Wang, Wen
• Is Part Of: Nonlinear analysis. Volume 214(2022)
• Publication Date: 2022-01
• Subjects: 58J35 -- 53C44; Hessian bound -- Gradient estimate -- Nonlinear parabolic equation -- Ricci flow -- Harnack inequality; 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.2021.112548
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In the paper, we first prove a Hamilton-Souplet-Zhang type gradient estimate for a positive solution to the nonlinear parabolic type equation ∂ t u ( x, t ) = ( Δ − q ( x, t ) ) u ( x, t ) + a u ( x, t ) log u ( x, t ) on Riemaniann manifolds along the Ricci flow. These estimates optimize the obtained conclusions by Bailesteanu et al. (2010) and Li and Zhu (2018). Secondly, by using Han-Zhang's method (Han and Zhang, 2016) and Hamilton-Souplet-Zhang type gradient estimate, we establish some global and local upper bounds for the Hessian of log positive solutions of the nonlinear parabolic type equations along the Ricci flow. As an application, we deduce some space-only Harnack type inequalities for bounded positive solutions of the parabolic type equation. Once more, by using Li's method (Li, 1991), we also derive some second order gradient estimates. Finally, we derive L 2 estimates for log-solutions to the parabolic type equation on Riemaniann manifolds.