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

$|{v}_{t}{|}^{p-2}{v}_{t}+{\left(-{\Delta }_{p}\right)}^{s}v=0,$

where

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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doubly nonlinear evolution, Hölder estimates, eigenvalue problem, fractional $p$-Laplacian, nonlocal equation