Cauchy problem of a nonlocal p-Laplacian evolution equation with nonlocal convection

This paper is concerned with the Cauchy problem of a nonlocal equation that takes into account convective and p-Laplacian diffusive effects∂u/ ∂t(x,t)=∫_RNJ(x-y)|u(y,t)-u(x,t)|^p-2(u(y,t)-u(x,t) )dy+(G*-f(u)-f(u))(x,t) with J radially symmetric and G not necessarily symmetric. First, we prove the existence and uniqueness of solutions, and if the convolution kernels J and G are rescaled appropriately, we show that solutions of the nonlocal problem converge to the solution of the usual p-Laplacian diffusion equation with convection. Finally, as a supplementary result, we study the asymptotic behavior of solutions as t → ∞ and give the decay estimate.