Long-time behavior for a nonlocal convection diffusion equation

In this paper we consider a nonlocal viscous Burgers' equation and study the well-posedness and asymptotic behavior of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers' equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.