Gradient estimates for Leibenson's equation on Riemannian manifolds

TL;DR

This paper derives gradient estimates for positive solutions of Leibenson's equation on Riemannian manifolds with Ricci curvature bounded below, distinguishing between slow and fast diffusion cases.

Contribution

It provides new gradient estimates for Leibenson's equation on Riemannian manifolds, considering different diffusion regimes and curvature conditions.

Findings

Gradient estimates established for solutions with Ricci curvature bounded below.

Different results obtained for slow and fast diffusion cases.

Enhances understanding of nonlinear PDE behavior on curved spaces.

Abstract

We consider on Riemannian manifolds solutions of the Leibenson equation \begin{equation*} \partial _{t}u=\Delta _{p}u^{q}. \end{equation*} This equation is also known as doubly nonlinear evolution equation. We prove gradient estimates for positive solutions $u$ under the condition that the Ricci curvature on $M$ is bounded from below by a non-positive constant. We distinguish between the case $q(p-1)>1$ (slow diffusion case) and the case $q(p-1)<1$ (fast diffusion case).