  The famous Fisher-KPP reaction diffusion model combines linear diffusion with
the typical Fisher-KPP reaction term, and appears in a number of relevant
applications. It is remarkable as a mathematical model since, in the case of
linear diffusion, it possesses a family of travelling waves that describe the
asymptotic behaviour of a wide class solutions $0\leq u(x,t)\leq 1$ of the
problem posed in the real line. The existence of propagation wave with finite
speed has been confirmed in the cases of "slow" and "pseudo-linear" doubly
nonlinear diffusion too, see arXiv:1601.05718. We investigate here the
corresponding theory with "fast" doubly nonlinear diffusion and we find that
general solutions show a non-TW asymptotic behaviour, and exponential
propagation in space for large times. Finally, we prove precise bounds for the
level sets of general solutions, even when we work in with spacial dimension $N
\geq 1$. In particular, we show that location of the level sets is
approximately linear for large times, when we take spatial logarithmic scale,
finding a strong departure from the linear case, in which appears the famous
Bramson logarithmic correction.
