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February  2010, 27(1): 301-323. doi: 10.3934/dcds.2010.27.301

On a nonlocal aggregation model with nonlinear diffusion

Dong Li 1, and  Xiaoyi Zhang 1,

1. 

Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242, United States, United States

Received  April 2009 Revised  December 2009 Published  February 2010

We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops $L_x^\infty$-norm blowup in finite time.
Keywords: nonlinear diffusion., Nonlocal, aggregation.
Mathematics Subject Classification: Primary: 35K59, 35K65; Secondary: 35Q9.
Citation: Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301
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