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

Authors

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)

Abstract

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 $u_0$.

Keywords

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

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Published 31 January 2017