  We consider the natural time-dependent fractional $p$-Laplacian equation
posed in the whole Euclidean space, with parameter $1<p<2$ and fractional
exponent $s\in (0,1)$. Rather standard theory shows that the Cauchy Problem for
data in the Lebesgue $L^q$ spaces is well posed, and the solutions form a
family of non-expansive semigroups with regularity and other interesting
properties. The superlinear case $p>2$ has been dealt with in a recent paper.
  We study here the "fast" regime $1<p<2$ which is more complex. As main
results, we construct the self-similar fundamental solution for every mass
value $M$ and any $p$ in the subrange $p_c=2N/(N+s)<p<2$, and we show that this
is the precise range where they can exist. We also prove that general
finite-mass solutions converge towards the fundamental solution having the same
mass, and convergence holds in all $L^q$ spaces. Fine bounds in the form of
global Harnack inequalities are obtained.
  Another main topic of the paper is the study of solutions having strong
singularities. We find a type of singular solution called Very Singular
Solution that exists for $p_c<p<p_1$, where $p_1$ is a new critical number that
we introduce, $p_1\in (p_c,2)$. We extend this type of singular solutions to
the "very fast range" $1<p<p_c$. They represent examples of weak solutions
having finite-time extinction in that lower $p$ range. We briefly examine the
situation in the limit case $p=p_c$. Finally, very singular solutions are
related to fractional elliptic problems of nonlinear eigenvalue form.in the
limit case $p=p_c$. Finally, very singular solutions are related to fractional
elliptic problems of nonlinear eigenvalue form.
