Large time behavior of solutions of Trudinger's equation

Abstract

We study the large time behavior of solutions $v:\Omega \times (0,\infty )\to \mathbb{R}$ of the PDE ${\partial }_t(|v{|}^{p-2}v)={\Delta }_{p}v.$ We show that $e^{\left({\lambda }_p/(p-1)\right)t}v(x,t)$ converges to an extremal of a Poincaré inequality on $\Omega$ with optimal constant ${\lambda }_p$, as $t\to \infty$. We also prove that the large time values of solutions approximate the extremals of a corresponding "dual" Poincaré 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.