Parabolic Gradient Estimates and Harnack Inequalities for a Nonlinear Equation Under The Ricci Flow

摘要

When the Riemannian metric evolves under the Ricci flow, we investigate parabolic gradient estimates (Li-Yau's type and J. Li's type) for positive solutions to the nonlinear parabolic equation (Delta - partial derivative(t))u = (p + 1) vertical bar del u vertical bar(2)/u + qu on the underlying manifold. Based on these gradient estimates, we derive associated Harnack inequalities, respectively.