Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 1

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

Pages: 47 – 77

DOI: https://dx.doi.org/10.4310/DPDE.2017.v14.n1.a4


Ciprian G. Gal (Department of Mathematics, Florida International University, Miami, Fl., U.S.A.)

Mahamadi Warma (Department of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, San Juan, Puerto Rico)


Let $\Omega \subset \mathbb{R}^N$ be an arbitrary bounded open set. We consider a degenerate parabolic equation associated to the fractional $p$-Laplace operator ${(- \Delta)}^s_p (p \geq 2 , s \in (0, 1))$ with the Dirichlet boundary condition and a monotone perturbation growing like ${\lvert \tau \rvert}^{q-2} \tau , q \gt p$ and with bad sign at infinity as ${\lvert \tau \rvert} \to \infty$. We show the existence of locally-defined strong solutions to the problem with any initial condition $u_0 \in L^r (\Omega)$ where $r \geq 2$ satisfies $r \gt N(q-p) / sp$. 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 $u_0$.


fractional $p$-Laplace operator, Dirichlet boundary conditions, degenerate non-linear parabolic equations, existence and regularity of local solutions, blow up

2010 Mathematics Subject Classification

35K55, 35K65, 35R11

Published 31 January 2017