  Aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis
with nonlinear diffusion are popular models for nonlocal aggregation phenomenon
and are a source of a number of interesting mathematical problems in nonlinear
PDE. The purpose of this work is twofold. First, we continue our previous work,
which focused on nonlocal aggregation, modeled with a convolution. The goal was
to unify the local and global theory of these convolution-type models,
including the identification of a sharp critical mass; however, some cases
involving unbounded domains were left open. In particular, the biologically
relevant case $\Real^2$ was not treated. In this paper, we present an
alternative proof of local existence, which now applies to $\Real^d$ for all $d
\geq 2$ and give global results that were left open. The proof departs from
previous work in that it uses a more direct and intuitive regularization that
constructs approximate solutions on $\Real^d$ instead of on sequences of
bounded domains. Second, this work develops the local, subcritical, and small
data critical theory for a variety of Patlak-Keller-Segel models with spatially
varying diffusion and decay rate of the chemo-attractant.
