Gradient estimate for the degenerate parabolic equation ut=ΔF(u)+H(u) on manifolds

In this paper, we study the Harnack differential inequality and the Hamilton-type gradient estimate for the positive solutions of the ecumenic degenerate parabolic equationut=Δ(F(u))+H(u) on a complete Riemannian manifold with F′(u)>0. We show that the Harnack quantity trick introduced by Li and Yau is still useful in our case, however, the arguments and computations are much more involved and skilled, and new Harnack quantities have to be constructed. Besides the Harnack differential inequality, we also derive the Hamilton-type gradient estimate with the help of the Harnack quantity trick, which improves the previous related work to more general cases. Some interesting applications of our new results are also presented.