DOI 10.1007/s00028-007-0377-9 The Neumann problem for nonlocal nonlinear diffusion equations

Abstract. We study nonlocal diffusion models of the form (γ (u))t (t, x) = J(x − y)(u(t, y) − u(t, x)) dy. Here is a bounded smooth domain and γ is a maximal monotone graph inR2. This is a nonlocal diffusion problem analogous with the usual Laplacian with Neumann boundary conditions. We prove existence and uniqueness of solutions with initial conditions in L1(). Moreover, when γ is a continuous function we find the asymptotic behaviour of the solutions, they converge as t → ∞ to the mean value of the initial condition. 1