Blow-up in multidimensional aggregation equations with mildly singular interaction kernels

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Abstract
We consider the multidimensional aggregation equation utr (urK u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lip- schitz or better). In the case of bounded initial data, nite time singularity has been proved for kernels with a Lipschitz point at the origin (6, 5), whereas for C2 kernels there is no nite-time blowup. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of nite time blowup that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally we conclude with a discussion of similarity solutions for the case K(x) =jxj and some open problems.

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