  We study the large time behavior of non-negative solutions to the nonlinear
diffusion equation with critical gradient absorption $$\partial\_t u -
\Delta\_{p}u + |\nabla u|^{q\_*} = 0 \quad \hbox{in}
(0,\infty)\times\mathbb{R}^N\ ,$$ for $p\in(2,\infty)$ and $q\_*:=p-N/(N+1)$.
We show that the asymptotic profile of compactly supported solutions is given
by a source-type self-similar solution of the $p$-Laplacian equation with
suitable logarithmic time and space scales. In the process, we also get optimal
decay rates for compactly supported solutions and optimal expansion rates for
their supports that strongly improve previous results.
