Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u^￠_n(t) = ?_{j ? mathbbZ^d} J_n-ju_j(t)-u_n(t)u^{prime}_{n}(t) = sum_{{jin}{{{mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? mathbbZ^dn in {mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?_{n ? mathbbZ^d}J_n = 1sum_{{n in {mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.