On a comparison principle for Trudinger's equation

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.