HOLDER ESTIMATES AND LARGE TIME BEHAVIOR FOR A NONLOCAL DOUBLY NONLINEAR EVOLUTION

The nonlinear and nonlocal PDE vertical bar nu(t)vertical bar(p-2) nu(t) + (-Delta p)(s)nu = 0, where (-Delta(p))(s) v(x,t) = 2 P.V. integral(Rn) vertical bar nu(x,t) - nu(x+y,t)vertical bar(p-2)nu(x,t)-nu(x,y,t))/vertical bar y vertical bar(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 Hlder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional p-Laplacian.