LAUSR

We study the comparison principle for non-negative solutions of the equation ∂ ⁡ ( | v | p - 2 ⁢ v ) ∂ ⁡ t = div ⁡ ( | ∇ ⁡ v | p - 2 ⁢ ∇ ⁡ v ) , 1 < p < ∞ . \frac{\partial(|v|^{p-2}v)}{\partial t}=\operatorname{div}(|\nabla v|^{p-2}% \nabla v),\quad 1 This equation is related to extremals of Poincaré inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions.