Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion

TL;DR

This paper unifies and extends the well-posedness theory for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, establishing local and global existence, uniqueness, and blow-up criteria across various dimensions.

Contribution

It generalizes the notion of criticality, computes critical masses, and proves well-posedness results for a broad class of models with degenerate diffusion.

Findings

Proved local well-posedness on bounded domains and in space for certain dimensions.

Established global well-posedness for subcritical problems.

Identified critical mass thresholds for finite-time blow-up and global existence.

Abstract

Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions $d\geq 2$ and in all of space for $d\geq 3$, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow up is possible for initial data of arbitrary mass.