Quantities Of Nonlinear Parabolic Equations On Riemannian Manifolds

In this thesis, we mainly study the quantities of solutions for several nonlin-ear parabolic equations on Riemannian manifolds, including the local and global Aronson-Benilan estimates of positive smooth solutions for weighted porous medium and weighted fast diffusion equations and the quantities of entropy for weighted porous medium equation, the global Hamilton gradient estimate of a positive bounded solution for certain nonlinear heat-type equation on closed Riemannian manifold and the global Li-Yau gradient estimates of a positive solution for the nonlinear heat-type equation on compact Riemannian manifold with nonconvex boundarv, a condition to extend mean curvature flow.More precisely, in the second chapter, we mainly study the local Li-Yau gra-dient estimates of positive smooth solutions for weighted porous medium and weighted fast diffusion equations and using the estimates to get Aronson-Benilan estimates and the quantities of entropy for weighted porous medium equation, and we get the ancient solutions for the two equations are constants under some curvature conditions. Our results extend the theorem in [53] by P. Lu, L. Ni, J. L. Vazquez and C. Villani, which is the Aronson-Benilan gradient estimates of smooth positive solutions for porous medium and fast diffusion equations, and also extend other quantities for porous medium and fast diffusion equations.In the third chapter, we get two gradient estimates of positive solution for a nonlinear heat type equation, ut-Δu = aulogu. One result is Hamilton gradient estimate of positive smooth solutions for the equation on closed manifold, and the result extend a theorem in [54] by L. Ma; and the other result is a global Li-Yau gradient estimate of positive solution for the equation on compact manifold with nonconvex boundary, the result extend a theorem in [69] by J. P. Wang.Finally, in the fourth chapter, we get a condition to extend mean curvature flow at the first singularity time, which the norm of the second fundamental form is bounded by the power of mean curvature and certain subcritical quantities concerning the mean curvature integral is bounded. Our result extend a theorem in [47] by N. Q. Le and N. Sesum and in [78] by H. W. Xu, F. Ye and E. T. Zhao.