Harnack inequality, heat kernel bounds and eigenvalue estimates under integral Ricci curvature bounds

Abstract

Let (Mn,gij) be a complete Riemannian manifold. We prove that for any p>n2, when k(p,1) is small enough, some parabolic type gradient bounds hold for the positive solutions of a nonlinear parabolic equationut=Δu+aulog⁡u, on geodesic balls B(O,r) in M with 0<r≤1. We can also derive the gradient estimates for any solutions to the above nonlinear parabolic equation along the Ricci flow on a closed manifold without any curvature conditions. As its application, we derive some parabolic type gradient estimates for a positive solution u(x,t) of the heat equation ut=Δu. Moreover, our estimate is stronger than Zhang and Zhang's estimate (see Remark 2 in Section 4). Those results are generalizations of Li-Yau, Hamilton, Li-Xu type gradient estimates under the integral Ricci curvature bounds.By utilizing the gradient estimates of the heat equation, we obtain Harnack inequalities, the upper bound and the lower bound for the heat kernel, eigenvalue estimate and the lower bound of Green's function on Riemannian manifolds under the integral Ricci curvature bounds.