LARGE TIME BEHAVIOR FOR A NONLOCAL NONLINEAR GRADIENT FLOW

We study the large time behavior of the nonlinear and nonlocal equation v(t) + (-Delta(p))(s) v = f, where p is an element of (1, 2) boolean OR (2, infinity), s is an element of (0, 1) and (-Delta(p))(s) v(x, t) = 2 P.V. integral(Rn) vertical bar v(x, t) - v(x + y, t)vertical bar(p-2)(v(x, t) - v(x + y, t))/vertical bar y vertical bar(n+sp) dy. This equation arises as a gradient flow in fractional Sobolev spaces. We obtain sharp decay estimates as t -> infinity. The proofs are based on an iteration method in the spirit of J. Moser previously used by P. Juutinen and P. Lindqvist.