A nonlocal convection-diffusion equation

In this paper we study a nonlocal equation that takes into account convective and diffusive effects, u(t) = J * u - u + G * (f(u)) - f(u) in R-d, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation u(t) = Delta u + b . del(f (u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t -> infinity when f (u) = |u|(q-1) u with q > 1. We find the decay rate and the first-order term in the asymptotic regime. (C) 2007 Elsevier Inc. All rights reserved.