Gradient Estimates For A Parabolic Equation With Negative Exponent On Riemannian Manifold

In this paper,we study gradient estimates for the positive solutions of a nonlinear parabolic equation with a negative exponent,on the complete non-compact Riemannian manifold M with N-Bakry-EmeryRicci curvature or ?-Bakry-Emery Ricci curvature.First,we introduce the background and status of domestic and international research on the gradient estimation on traditional Riemannian manifolds,Bakry-Emery Ricci curvature and the gradient estimation on Riemannian manifolds with Bakry-Emery Ricci curvature.Then,we introduce the preliminary knowledge required to prove the main conclusions of the article and the Bochner formula and Bochner formula for N-Bakry-Emery Ricci curvature and ?-Bakry-Emery Ricci curvature ard proved.Finally,we discuss the gradient estimates for the positive solutions of a nonlinear parabolic equation with a negative exponent on the complete non-compact Riemannian manifold M with N-Bakry-EmeryRicci curvature or ?-Bakry-Emery Ricci curvature.Some propositions and corollary are proved.