Sharp sub-Gaussian upper bounds for subsolutions of Trudinger's equation on Riemannian manifolds

TL;DR

This paper establishes sharp sub-Gaussian upper bounds for weak subsolutions of Trudinger's nonlinear parabolic equation on Riemannian manifolds, including Euclidean space, advancing understanding of their behavior.

Contribution

It provides the first sharp sub-Gaussian upper bounds for subsolutions of Trudinger's equation on Riemannian manifolds, including Euclidean space.

Findings

Weak subsolutions have sub-Gaussian upper bounds.

The bounds are proven to be sharp for certain manifolds.

Results extend understanding of nonlinear parabolic equations on manifolds.

Abstract

We consider on Riemannian manifolds the nonlinear evolution equation \begin{equation*} \partial _{t}u=\Delta _{p}(u^{1/(p-1)}), \end{equation*}% where $p>1$. This equation is also known as a doubly non-linear parabolic equation or Trudinger's equation. We prove that weak subsolutions of this equation have a sub-Gaussian upper bound and prove that this upper bound is sharp for a specific class of manifolds including $\mathbb{R}^{n}$.