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Aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxiswith nonlinear diffusion are popular models for nonlocal aggregation phenomenonand are a source of a number of interesting mathematical problems in nonlinearPDE. The purpose of this work is twofold. First, we continue our previous work,which focused on nonlocal aggregation, modeled with a convolution. The goal wasto unify the local and global theory of these convolution-type models,including the identification of a sharp critical mass; however, some casesinvolving unbounded domains were left open. In particular, the biologicallyrelevant case $\Real^2$ was not treated. In this paper, we present analternative 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 fromprevious work in that it uses a more direct and intuitive regularization thatconstructs approximate solutions on $\Real^d$ instead of on sequences ofbounded domains. Second, this work develops the local, subcritical, and smalldata critical theory for a variety of Patlak-Keller-Segel models with spatiallyvarying diffusion and decay rate of the chemo-attractant.

Inhomogeneous Patlak-Keller-Segel models and Aggregation Equations with Nonlinear Diffusion in $\Real^d$