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• W2035731630 abstract "In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K * ρ) in documentclass[12pt]{minimal}begin{document}$mathbb {R}^d$end{document}Rd, d ⩾ 2, where K(r) = rγ/γ with γ &gt; 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math. 70, 2582–2603 (2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ &gt; 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 &lt; γ &lt; 4. For γ &gt; 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two-dimensional (in-)stability implies n-dimensional (in-)stability." @default.
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• W2035731630 date "2012-09-06" @default.
• W2035731630 modified "2023-10-16" @default.
• W2035731630 title "Stability and clustering of self-similar solutions of aggregation equations" @default.
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• W2035731630 doi "https://doi.org/10.1063/1.4745180" @default.
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