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

Jacob Bedrossian; Nancy Rodríguez

doi:10.3934/dcdsb.2014.19.1279

Article Contents

2014, Volume 19, Issue 5: 1279-1309. Doi: 10.3934/dcdsb.2014.19.1279

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

Jacob Bedrossian^1, and
Nancy Rodríguez^2,

1.
New York University, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012-1185
2.
Stanford University, Department of Mathematics, Building 380, Sloan Hall Stanford, CA 94305

Received: September 2011

Revised: March 2012

Published: July 2014

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Abstract

Aggregation equations and parabolic-elliptic Patlak-Keller-Segel (PKS) systems for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation 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.

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Mathematics Subject Classification: Primary: 35K55, 35K15; Secondary: 35K65.

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