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

Abstract

We consider a degenerate parabolic equation associated with the fractional -Laplace operator ${\left(-\Delta \right)}_p^s$\ ($p\ge 2$, $s\in \left(0,1\right)$) and a monotone perturbation growing like ${\left|s\right|}^{q-2}s,$ $q>p$ and with bad sign at infinity as $\left|s\right|\to \infty$. We show the existence of locally-defined strong solutions to the problem with any initial condition $u_0\in L^r(\Omega )$ where $r\ge 2$ 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 $u_0$.

arXiv comment31 pages in Dynamics of Partial Differential Equations 2016