Upper bounds for solutions of Leibenson's equation on Riemannian manifolds

Abstract

We consider on Riemannian manifolds the Leibenson equation${\partial }_{t}u={\Delta }_p{u}^q$that is also known as a doubly nonlinear evolution equation. We prove upper estimates of weak subsolutions to this equation on Riemannian manifolds with non-negative Ricci curvature in the case when p and q satisfy the conditions$1<p<2\text{and}1\le q<\frac{1}{p-1}.$ We show that these estimates are optimal in terms of long time behavior and near-optimal in terms of long distance behavior.