Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$

Abstract

Aggregation equations and parabolic-elliptic Patlak-Keller-Segel (PKS) systems for chemotaxis with nonlinear diffusion are popular models for nonlocalaggregation phenomenon and are a source of many interesting mathematical problems in nonlinear PDEs.The purpose of this work is to give a more complete study of local, subcritical and small-data critical/supercritical theory in $\mathbb{R}^d$, $d \geq 2$.Some existing results can be found in the literature; however, one of the most important cases in biological applications, that is the $\mathbb{R}^2$ case, had not been studied.In this paper, we treat two related systems, which are different generalizations of the classical parabolic-elliptic PKS model.In the first class, nonlocal aggregation is modeled by convolution with a general interaction potential, studied in this generality in our previous work [6].For this class of models we also present several large data global existence results for critical problems.The second class is a variety of PKS models with spatially inhomogeneous diffusion and decay rate of the chemo-attractant, which is potentially relevant to biological applications and raises interesting mathematical questions.