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.