Gradient Estimates Of Parabolic Equation On Riemannian Manifold

In this paper, we consider two problems: The first one is the gradient estimates forpositive solutions to the following nonlinear parabolic equationut=fu+au log u+buon Mn, where a, b are two constants. The fis the Witten Laplacian associated withf∈C∞(Mn); The second one is the gradient estimates for positive harmonic functionsof the Witten Laplacian.In chapter one, we introduce some developments of gradient estimates and relevantknowledge. For example, we introduce the concepts of Bochner formular, N-Bakry-EmeryRicci tensor RicNfand∞-Bakry-Emery Ricci tensor Ricf; Finally, we put forward thequestions considered in the paper.In chapter two, we study two kinds of problems. When N-Bakry-Emery Ricci tensorRicfN≥(n1)K and∞-Bakry-Emery Ricci tensor Ricf≥(n1)K, K≥0, weobtain gradient estimates for positive solutions of the above nonliner parabolic equation.In chapter three, we keep on studying gradient estimates for positive solutions of theabove nonliner parabolic equation and obtain gradient estimates of Hamiltone-type.In the last chapter, we consider gradient estimates for positive harmonic functionsof the Witten Laplacian. When the∞-Bakry-Emery Ricci tensor is bounded from belowand|f|is bounded, we obtain a Liouvill-type theorem. This extends a result of Cheng-Yau.