Journal
NONLINEARITY
Volume 22, Issue 3, Pages 683-710Publisher
IOP PUBLISHING LTD
DOI: 10.1088/0951-7715/22/3/009
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Funding
- ARO [W911NF-05-1-0112]
- ONR [N000140610059]
- DGI-MICINNC [MTM2008-06349C03-03]
- ICREA Funding Source: Custom
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We consider the multidimensional aggregation equation u(t) - del. (u del K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717-35), whereas for C-2 kernels there is no finite-time blow-up. 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 finite time blow-up 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) = vertical bar x vertical bar and some open problems.
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