Li-Yau Estimates For Some Parabolic And Ultraparabolic Equations

In recent years,many experts and scholars have done a lot of researches on the heat equation and its extensions,and obtained Li-Yau Harnack estimate(differential Harnack estimate)and Li-Yau gradient estimate of positive solutions.These equations play important roles in describing processes in biology and other fields.For example,Castaway equation,Endangered species equation,Kolmogorov equation,porous medium equation,parabolic equation on manifold and so on.In this paper,we study Li-Yau Harnack estimate and Li-Yau gradient estimate of parabolic and ultraparabolic equations.Firstly,we prove the matrix Li-Yau Harnack estimate for a class of Kolmogorov type equations.Using the maximum principle,we prove the matrix Li-Yau Harnack estimate,and use this Li-Yau Harnack estimate to select the optimal path to give the Harnack inequality for the positive solution at two points in different time and space.Secondly,Li-Yau estimate of a kind of porous medium equations is discussed,and the properties of the solution 1)and log 1)are studied respectively.Two different Li-Yau Harnack estimates are obtained,and then two different relations of the solutions at two points in different time and space are obtained.Finally,Li-Yau type gradient estimates for parabolic equations on a complete Riemannian manifold are investigated,and Harnack inequalities and the solutions' blow-up properties in finite time are obtained according to Li-Yau gradient estimates.