The Dirichlet problem for the fractional p-Laplacian evolution equation

We consider a model of fractional diffusion involving a natural nonlocal version of the p-Laplacian operator. We study the Dirichlet problem posed in a bounded domain Omega of R-N with zero data outside of Omega, for which the existence and uniqueness of strong nonnegative solutions is proved, and a number of quantitative properties are established. A main objective is proving the existence of a special separate variable solution U(x, t) = t(-1/(p-2)) F(x), called the friendly giant, which produces a universal upper bound and explains the large-time behaviour of all nontrivial nonnegative solutions in a sharp way. Moreover, the spatial profile F of this solution solves an interesting nonlocal elliptic problem. We also prove everywhere positivity of nonnegative solutions with any nontrivial data, a property that separates this equation from the standard p-Laplacian equation. (C) 2015 Elsevier Inc. All rights reserved.