  We study the propagation properties of nonnegative and bounded solutions of
the class of reaction-diffusion equations with nonlinear fractional diffusion:
$u_{t} + (-\Delta)^s (u^m)=f(u)$. For all $0<s<1$ and $m> m_c=(N-2s)_+/N $, we
consider the solution of the initial-value problem with initial data having
fast decay at infinity and prove that its level sets propagate exponentially
fast in time, in contradiction to the traveling wave behaviour of the standard
KPP case, which corresponds to putting $s=1$, $m=1$ and $f(u)=u(1-u)$. The
proof of this fact uses as an essential ingredient the recently established
decay properties of the self-similar solutions of the purely diffusive
equation, $u_{t} + (-\Delta)^s u^m=0$.
