  In this paper we consider a subcritical model that involves nonlocal
diffusion and a classical convective term. In spite of the nonlocal diffusion,
we obtain an Oleinik type estimate similar to the case when the diffusion is
local. First we prove that the entropy solution can be obtained by adding a
small viscous term $\mu u_{xx}$ and letting $\mu\to 0$. Then, by using uniform
Oleinik estimates for the viscous approximation we are able to prove the
well-posedness of the entropy solutions with $L^1$-initial data. Using a
scaling argument and hyperbolic estimates given by Oleinik's inequality, we
obtain the first term in the asymptotic behavior of the nonnegative solutions.
Finally, the large time behavior of changing sign solutions is proved using the
classical flux-entropy method and estimates for the nonlocal operator.
