Upper bounds of Hessian matrix and gradient estimates of positive solutions to the nonlinear parabolic equation along Ricci flow

Abstract

In the paper, we first prove a Hamilton-Souplet-Zhang type gradient estimate for a positive solution to the nonlinear parabolic type equation ∂tu(x,t)=(Δ−q(x,t))u(x,t)+au(x,t)logu(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 L2 estimates for log-solutions to the parabolic type equation on Riemaniann manifolds.