On a nonlocal aggregation model with nonlinear diffusion

Dong Li; Xiaoyi Zhang

doi:10.3934/dcds.2010.27.301

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2010, Volume 27, Issue 1: 301-323. Doi: 10.3934/dcds.2010.27.301

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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

Received: March 31, 2009

Revised: November 30, 2009

Published: February 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K59, 35K65; Secondary: 35Q92.

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