Fractional p-Laplacian evolution equations

In this paper we study the fractional p-Laplacian evolution equation given by u(t) (t, x) = integral(A) 1/vertical bar x - y vertical bar N+ sp vertical bar u(t, y) - u (t, x)vertical bar(p-2)(u(t, y) - u(t, x) )dy for x is an element of Omega, t > 0, 0 < s < 1, p >= 1. In a bounded domain Omega we deal with the Dirichlet problem by taking A = R-N and u = 0 in R-N \ Omega, and the Neumann problem by taking A = Omega. We include here the limit case p = 1 that has the extra difficulty of giving a meaning to u(y)-u(x)/vertical bar u(v)-u(x) when u(y) = u(x). We also consider the Cauchy problem in the whole R-N by taking A = Omega = R-N. We find existence and uniqueness of strong solutions for each of the above mentioned problems. We also study the asymptotic behaviour of these solutions as t -> infinity. Finally, we recover the local p-Laplacian evolution equation with Dirichlet or Neumann boundary conditions, and for the Cauchy problem, by taking the limit as s -> 1 in the nonlocal problems multiplied by a suitable scaling constant. (C) 2016 Elsevier Masson SAS. All rights reserved.