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

Author: Bedrossian Jacob   Rodríguez Nancy   Bertozzi Andrea L  

Publisher: IOP Publishing

ISSN: 0951-7715

Source: Nonlinearity, Vol.24, Iss.6, 2011-06, pp. : 1683-1714

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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.