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

TL;DR

This paper studies a degenerate non-local parabolic equation involving the fractional p-Laplacian, establishing existence of solutions and conditions under which solutions can blow up in finite time.

Contribution

It introduces new existence results for strong solutions with general initial data and identifies parameter regimes leading to finite time blow-up.

Findings

Existence of locally-defined strong solutions for initial data in L^r()

Finite time blow-up can occur depending on parameters and initial conditions

Conditions relating r, p, q, and initial data for blow-up are established

Abstract

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 $u_0$.