Vol. 9, No. 6, 2016

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Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution

Ryan Hynd and Erik Lindgren

Vol. 9 (2016), No. 6, 1447–1482
Abstract

The nonlinear and nonlocal PDE

|vt|p2v t + (Δp)sv = 0,

where

(Δp)sv(x,t) = 2  P.V.n|v(x,t) v(x + y,t)|p2(v(x,t) v(x + y,t)) |y|n+sp dy,

has the interesting feature that an associated Rayleigh quotient is nonincreasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide Hölder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional p-Laplacian.

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Keywords
doubly nonlinear evolution, Hölder estimates, eigenvalue problem, fractional $p$-Laplacian, nonlocal equation
Mathematical Subject Classification 2010
Primary: 35J60, 47J35, 35J70, 35R09
Milestones
Received: 18 November 2015
Revised: 16 May 2016
Accepted: 17 June 2016
Published: 3 October 2016
Authors
Ryan Hynd
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104
United States
Erik Lindgren
Department of Mathematics
Royal Institute of Technology
100 44 Stockholm
Sweden