  We consider a degenerate parabolic equation associated with the fractional $%
p $-Laplace operator $\left( -\Delta \right) _{p}^{s}$\ ($p\geq 2$, $s\in
\left( 0,1\right) $) and a monotone perturbation growing like $\left\vert
s\right\vert ^{q-2}s,$ $q>p$ and with bad sign at infinity as $\left\vert
s\right\vert \rightarrow \infty $. We show the existence of locally-defined
strong solutions to the problem with any initial condition $u_{0}\in
L^{r}(\Omega )$ where $r\geq 2$ satisfies $r>N(q-p)/sp$. Then, we prove that
finite time blow-up is possible for these problems in the range of parameters
provided for $r,p,q$ and the initial datum $u_0$.
