We study the large time behavior of nonlinear and nonlocal equation \begin{document}$ v_t+(- \Delta_p)^sv = f \, , $\end{document} where $ p\in (1, 2)\cup (2, \infty) $, s\in (0, 1) and$ (- \Delta_p)^s v\, (x, t) 2 {\rm{P.V.}} \int_{ \mathbb{R}^n}\frac{|v(x, t)-v(x+y, t)|^{p-2}(v(x, t))}{|y|^{n+sp}}\, dy. $This arises as a gradient flow in fractional Sobolev spaces. obtain sharp decay estimates...