November  2010, 9(6): 1591-1606. doi: 10.3934/cpaa.2010.9.1591

Global wellposedness and blowup of solutions to a nonlocal evolution problem with singular kernels

1. 

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

Received  October 2009 Revised  February 2010 Published  August 2010

We consider a nonlocal evolution equation in $R^2$: $\partial_t u + \nabla \cdot (u K*u )= 0$, where $K(x) = \mu \frac x {|x|^\alpha}$, $\mu=\pm 1$ and $1 < \alpha < 2 $. We study wellposedness, continuation/blowup criteria and smoothness of solutions in Sobolev spaces. In the repulsive case ($\mu=1$), by using the sharp blowup criteria, we prove global wellposedness for any positive large initial data. In the attractive case ($\mu=-1$), by using a novel free energy inequality together with a mass localization technique, we construct finite time blowups for a large class of smooth initial data.
Citation: Dong Li, Xiaoyi Zhang. Global wellposedness and blowup of solutions to a nonlocal evolution problem with singular kernels. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1591-1606. doi: 10.3934/cpaa.2010.9.1591
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