# American Institute of Mathematical Sciences

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