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

Abstract

The nonlinear and nonlocal PDE

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

where

$${\left(-{\Delta }_p\right)}^{s}v\left(x,t\right)=2\text{P.V.}{\int }_{{ℝ}^n}\frac{|v\left(x,t\right)-v\left(x+y,t\right){|}^{p-2}\left(v\left(x,t\right)-v\left(x+y,t\right)\right)}{|y{|}^{n+sp}}dy,$$

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.