Abstract
We consider an aggregation equation in \({\mathbb {R}^n}\), n ≥ 2 with fractional dissipation, namely, \({u_t + \nabla\cdot(u \nabla K*u)=-\nu (-\Delta)^{\gamma/2} u}\), where 0 ≤ γ < 1 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e −|x|. We prove that for a class of smooth initial data, the solutions develop blow-up in finite time.
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Li, D., Rodrigo, J. Finite-Time Singularities of an Aggregation Equation in \({\mathbb {R}^n}\) with Fractional Dissipation. Commun. Math. Phys. 287, 687–703 (2009). https://doi.org/10.1007/s00220-008-0669-0
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DOI: https://doi.org/10.1007/s00220-008-0669-0