  We consider the natural time-dependent fractional $p$-Laplacian equation
posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$
(fractional exponent). We show that the Cauchy Problem for data in the Lebesgue
$L^q$ spaces is well posed, and show that the solutions form a family of
non-expansive semigroups with regularity and other interesting properties. As
main results, we construct the self-similar fundamental solution for every mass
value $M,$ and prove that general finite-mass solutions converge towards that
fundamental solution having the same mass in all $L^q$ spaces.A number of
additional properties and estimates complete the picture.
