  We study the large time behavior of the nonlinear and nonlocal equation $$
v_t+(-\Delta_p)^sv=f \, , $$ where $p\in (1,2)\cup (2,\infty)$, $s\in (0,1)$
and $$ (-\Delta_p)^s v\, (x,t)=2 \,\text{pv}
\int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\,
dy. $$ This equation arises as a gradient flow in fractional Sobolev spaces. We
obtain sharp decay estimates as $t\to\infty$. The proofs are based on an
iteration method in the spirit of J. Moser previously used by P. Juutinen and
P. Lindqvist.
