On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian

We consider a degenerate parabolic equation associated with the fractional $% p $-Laplaceoperator$\left( -\Delta \right) {p}^{s}$($p\geq 2$,$s\in \left( 0,1\right) $)andamonotoneperturbationgrowinglike$\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 $u_0$.