Long time upper bounds for solutions of Leibenson's equation on Riemannian manifolds

TL;DR

This paper establishes that an upper bound condition for weak solutions of Leibenson's equation on Riemannian manifolds is equivalent to a Sobolev inequality of Euclidean type, linking PDE behavior to geometric inequalities.

Contribution

It demonstrates the equivalence between solution bounds for Leibenson's equation and Sobolev inequalities on Riemannian manifolds, extending Euclidean results to curved spaces.

Findings

Upper bounds for solutions are equivalent to Sobolev inequalities.

The results connect PDE estimates with geometric inequalities.

The work generalizes Euclidean PDE bounds to Riemannian manifolds.

Abstract

We consider on Riemannian manifolds the Leibenson equation $$\partial _{t}u=\Delta _{p}u^{q}.$$ We prove that a certain upper bound for weak solutions of this equation is equivalent to a euclidean-type Sobolev inequality.