# Large time behavior of solutions of Trudinger's equation

## TL;DR

This paper analyzes the long-term behavior of solutions to Trudinger's equation, showing convergence to extremals of Poincaré inequalities and providing asymptotics for related nonlinear flows with various boundary conditions.

## Contribution

It establishes the exponential convergence of solutions to extremals of Poincaré inequalities and extends the analysis to related nonlinear flows with different boundary conditions.

## Key findings

- Solutions scaled by an exponential factor converge to Poincaré extremals.
- Large time solutions approximate dual Poincaré extremals.
- Results apply to various boundary conditions and nonlocal operators.

## Abstract

We study the large time behavior of solutions $v:\Omega\times(0,\infty)\rightarrow \mathbb{R}$ of the PDE $\partial_t(|v|^{p-2}v)=\Delta_pv.$ We show that $e^{\left(\lambda_p/(p-1)\right)t}v(x,t)$ converges to an extremal of a Poincar\'e inequality on $\Omega$ with optimal constant $\lambda_p$, as $t\rightarrow \infty$. We also prove that the large time values of solutions approximate the extremals of a corresponding "dual" Poincar\'e inequality on $\Omega$. Moreover, our theory allows us to deduce the large time asymptotics of related doubly nonlinear flows involving various boundary conditions and nonlocal operators.