Abstract: | In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491 ), we shall consider the nonlinear parabolic equation on Riemannian manifolds with , . First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when , our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang. |