Large time behavior of solutions of Trudinger's equation

We study the large time behavior of solutions nu : Omega x (0, infinity) -> R of the PDE partial derivative(t)(vertical bar nu vertical bar(p-2)nu) = Delta(p)nu. We show that e((lambda p/(p-1))t) nu(x, t) converges to an extremal of a Poincare inequality on Omega with optimal constant lambda(p), as t -> infinity. We also prove that the large time values of solutions approximate the extremals of a corresponding "dual" Poincare 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. (C) 2020 Elsevier Inc. All rights reserved.