Cauchy problem of a nonlocal [formula omitted]-Laplacian evolution equation with nonlocal convection

Abstract

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.