Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion

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 ⩾ 2 and in all of space for d ⩾ 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.