Local Gradient Estimates Of Aronson-Bénilan Type Nonlinear Equations On Riemannian Manifold

In this work we research the followed Aronson-Benilan type nonlinear equations posed on Riemannian manifolds with a lower Ricci curvature bound.(?)fu=△um+bu.On account of the work about porous medium equations posed on Riemannian manifolds in[14],following are local gradient estimate that we derive in this paper for positive solutions of the equation above.Let u be a positive smooth solution to equation (?)tu=△um+bu,m>1,and Q:=B(O,R)×[O,T],Let vmaxR,T:=maxB(O,R)×[O,T]v.(1)Assume that the Ricci curvature Ricci≥0on B(O,R).Then,for any α>1we have: on Q’:=B(O,R)×[O,T].Here b is a constant that depend on m, C1:=40(m-1)(n+2), C2(α):=(200aα2m2)/(α-1),a:=(n(m-1))/(n(m-1)+2).(2)Assume that Ricci≥-(n-1)K2on B(O,R)for some K≥0.Then,for any α>1,we have,on Q’, Here C6:=(200aα2m2)/(α-1)/(m-1)[80+40((n-1)(1+KR)+1)].