Nonlocal diffusion equations

In this paper we present a variational approach to establish the existence and uniqueness of variational solutions to nonlocal evolutionary problems. The model we have in mind is the following nonlocal version of the parabolic p-Laplacianut−div[a(‖Du(t)‖Lp(Ω)p)|Du|p−2Du]=h, which has recently attracted many authors. We emphasize that our approach also applies in situations where the considered functionals do not allow the derivation of the associated Euler–Lagrange equation.