Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Asymptotics of blowup solutions for the aggregation equation

Pages: 1309 - 1331, Volume 17, Issue 4, June 2012      doi:10.3934/dcdsb.2012.17.1309

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Yanghong Huang - Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada (email)
Andrea Bertozzi - 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, United States (email)

Abstract: We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation $ u_t = \nabla\cdot(u\nabla K*u) $ in $\mathbb{R}^n$, for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$. For $\gamma>2$, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing $\delta$-ring. We develop an asymptotic theory for the approach to this singular solution. For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all $\gamma$ in this range, including additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.

Keywords:  Aggregation equation, blowup, asymptotic behavior, self-similar solutions.
Mathematics Subject Classification:  Primary: 35B40, 35B44; Secondary: 92D50.

Received: January 2011;      Revised: September 2011;      Available Online: February 2012.