Nonlocal $p$-Laplace equations depending on the $L^p$ norm of the gradient

Abstract

We are studying a class of nonlinear nonlocal diffusion problems associated with a p-Laplace-type operator, where a nonlocal quantity is present in the diffusion coefficient. We address the issues of existence and uniqueness for the parabolic setting. Then, we study the asymptotic behavior of the solution for large time. For this purpose, we introduce and investigate, in detail, the associated stationary problem. Moreover, since the solutions of the stationary problem are also critical points of some energy functional, we make a classification of its critical points.