Sharp long distance upper bounds for solutions of Leibenson's equation on Riemannian manifolds

TL;DR

This paper establishes sharp upper bounds for weak subsolutions of Leibenson's equation on Riemannian manifolds with non-negative Ricci curvature, covering all relevant parameters and confirming a previous conjecture.

Contribution

It improves previous results by providing optimal upper estimates for solutions of Leibenson's equation across the entire parameter range on specific Riemannian manifolds.

Findings

Proved sharp upper bounds for weak subsolutions on manifolds with non-negative Ricci curvature.

Extended the range of parameters for which bounds are valid.

Confirmed a conjecture from prior work by Grigor'yan et al.

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 sharp upper estimates of weak subsolutions to this equation on Riemannian manifolds with non-negative Ricci curvature in the whole range of $p>1$ and $q>0$ satisfying $q(p-1)<1$. In this way, we improve the result of \cite{Grigoryan2024a} and prove Conjecture 1.2 from \cite{Grigoryan2024a}.