Harnack Estimate Of One Class Of Aronson-Benilan Type Nonlinear Equations

In this thesis,we consider two problems:The first one is about the following Aronson-Benilan type nonlinear equation (?)tu=△φup+bu,which is an extension under Witten Lalacian,where b,p are two constants,△φ=-▽φ.▽；The second one is about the gradient estimate of an Aronson-Benilan type nonlinear equation posed on Riemannian manifolds when the metric is evolved by Ricci flow.On account of the work about porous medium equations posed on Riemannian mani-folds in[2],following are local gradient estimate that we derive in this paper for positive solutions of the equation above.(1)Mn is a complete Riemannian manifold,Ricmφ(Bp(2R))≥-K,K≥0.Let u be a positive smooth solution to equation (?)tu=△φup+bu,and p>1.Let v,=p/p-1up-1M=(p-1)maxBp(2R)×[0,T]v.Then,for any α>1,on the geodesic ball Bp(R),we have:(2)Let(Mn,g(t)),t∈[0,T] be a solution to the Ricci flow (?)/(?)tg=-2Rc on a closed n-manifo1d.Assume that|Rc|≤k,let u be a positive smooth solution to equation (?)u/(?)t=△gum+bu,v=p/p-1up-1,M=(p-1)maxBp(2R)×[0,T]v.Then,for any α>1,on the geodesic ball,we have:...