Differential Harnack estimates for a weighted nonlinear parabolic equation under a super Perelman-Ricci flow and implications

In this paper, we derive new differential Harnack estimates of Li-Yau type forpositive smooth solutions to a class of nonlinear parabolic equations in the formLa phi[w]:=[partial derivative partial derivative t-a(x, t)-Delta phi]w(x, t)=G(t, x, w(x, t)),t>0,on smooth metric measure spaces where the metric and potential are time dependentand evolve under a (k,m)-super Perelman-Ricci flow. A number of consequences,most notably, a parabolic Harnack inequality, a class of Hamilton type globalcurvature-free estimates and a general Liouville type theorem together with someconsequences are established. Some special cases are presented to illustrate thestrength of the resultss are established. Some special cases are presented to illustrate the strength of the results.

Automated summary

In this paper, new differential Harnack estimates of Li-Yau type are derived for positive smooth solutions to a class of nonlinear parabolic equations. The equations are in the form La phi[w]:=[partial derivative partial derivative t-a(x, t)-Delta phi]w(x, t)=G(t, x, w(x, t)),t>0, on smooth metric measure spaces where the metric and potential are time dependent and evolve under a (k,m)-super Perelman-Ricci flow. Several consequences, including a parabolic Harnack inequality, a class of Hamilton type global curvature-free estimates, and a general Liouville type theorem, along with some consequences, are established. Some special cases are presented to illustrate the strength of the results.