Gradient estimates for $u_t=\Delta F(u)$ on manifolds and some Liouville-type theorems

Abstract

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=\Delta F(u),$$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): $$u_t=\Delta (u^p),\qquad p>0,$$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of $p$ than that for Aronson-B\'enilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yaus celebrated Liouville theorem for positive harmonic functions.