Differential Harnack Estimates Of Solutions Of A Class Of Nonlinear Parabolic Equations (Systems

In 1986,Li and Yau got a differential Harnack estimate for positive solutions to heat equation on a Riemann manifold with nonnegative Ricci curvature,and this estimate is optimal.Since then,experts and scholars have extended this estimate to linear and nonlinear parabolic equations on manifolds.Inspired by these,we study differential Harnack estimates for positive solutions to a kind of nonlinear parabolic system and a kind of nonlinear parabolic equation in this paper.Firstly,we consider a nonlinear parabolic system(?) on N-dimensional Euclidean space.We prove a differential Harnack estimate for positive solutions by maximum principle.We derive classical Harnack inequalities by this estimate,and we show that positive solutions to this equation blow up in finite time.Secondly,We study a nonlinear parabolic equation f_t=Δ_H~Nf+p(t)f~m+q(t)f~n on N-dimensional hyperbolic space.We derive a differential Harnack estimate for positive solutions to this equation through the same method.As applications,we derive a classical Hsarnack inequality and the property of blow-up solutions.