Li-Yau Estimates and Harnack Inequalities for Nonlinear Slow Diffusion Equations on a Smooth Metric Measure Space

We present new gradient estimates and Harnack inequalities for positive solutions to nonlinear slow diffusion equations. The framework is that of a smooth metric measure space $(\mathscr M,g,d\mu)$ with invariant weighted measure $d\mu=e^{-\phi} dv_g$ and diffusion operator $\Delta_\phi=e^\phi {\rm div} (e^{-\phi} \nabla)$ -- the $\phi$-Laplacian. The nonlinear slow diffusion equation, then, for $x \in {\mathscr M}$ and $t>0$, and fixed exponent $p>1$, takes the form \begin{equation*} \partial_t u (x,t) - \Delta_\phi u^p (x,t) = \mathscr N (t,x,u(x,t)). \end{equation*} We assume that the metric tensor $g$ and potential $\phi$ are space-time dependent; hence the same is true of the usual metric and potential dependent differential operators and curvature tensors. The estimates are established under natural lower bounds on the Bakry-\'Emery $m$-Ricci curvature tensor and the time derivative of metric tensor. The curious interplay between geometry, nonlinearity and evolution and their influence on the estimates is at the centre of this investigation. The results here considerably extend and improve earlier results on slow diffusion equations. Several implication, special cases and corollaries are presented and discussed.