# On a comparison principle for Trudinger's equation

## TL;DR

This paper investigates the comparison principle for solutions of a nonlinear PDE related to Sobolev inequalities, providing insights into the long-term behavior of solutions.

## Contribution

It establishes a comparison principle for Trudinger's equation and applies it to analyze the asymptotic behavior of solutions over time.

## Key findings

- Established a comparison principle for non-negative solutions.
- Derived pointwise estimates for large time behavior.
- Connected the PDE analysis to Sobolev inequality extremals.

## Abstract

We study the comparison principle for non-negative solutions of the equation $$ \frac{\partial\,(|v|^{p-2}v)}{\partial t}\,=\, \textrm{div} (|\nabla v|^{p-2}\nabla v), \quad 1<p<\infty.$$ This equation is related to extremals of Poincar\'e inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions.