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

This paper is concerned with the Cauchy problem of a nonlocal equation that takes into account convective and pp-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 JJ radially symmetric and GG not necessarily symmetric. First, we prove the existence and uniqueness of solutions, and if the convolution kernels JJ and GG are rescaled appropriately, we show that solutions of the nonlocal problem converge to the solution of the usual pp-Laplacian diffusion equation with convection. Finally, as a supplementary result, we study the asymptotic behavior of solutions as t→∞t→∞ and give the decay estimate.