  We consider a class of time dependent second order partial differential
equations governed by a decaying entropy. The solution usually corresponds to a
density distribution, hence positivity (non-negativity) is expected. This class
of problems covers important cases such as Fokker-Planck type equations and
aggregation models, which have been studied intensively in the past decades. In
this paper, we design a high order discontinuous Galerkin method for such
problems. If the interaction potential is not involved, or the interaction is
defined by a smooth kernel, our semi-discrete scheme admits an entropy
inequality on the discrete level. Furthermore, by applying the
positivity-preserving limiter, our fully discretized scheme produces
non-negative solutions for all cases under a time step constraint. Our method
also applies to two dimensional problems on Cartesian meshes. Numerical
examples are given to confirm the high order accuracy for smooth test cases and
to demonstrate the effectiveness for preserving long time asymptotics.
