  We analyze a Discontinuous Galerkin method for a problem with linear
advection-reaction and $p$-type diffusion, with Sobolev indices $p\in (1,
\infty)$. The discretization of the diffusion term is based on the full
gradient including jump liftings and interior-penalty stabilization while, for
the advective contribution, we consider a strengthened version of the classical
upwind scheme. The developed error estimates track the dependence of the local
contributions to the error on local P\'eclet numbers. A set of numerical tests
supports the theoretical derivations.
