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Mathematics  2012 

Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2

DOI: 10.1007/s10440-013-9832-5

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Abstract:

In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned lack of non-decay assumption is essential with respect to keeping the dichotomy finite-time blowup against boundedness of solutions. Namely, we prove that without the non-decay assumption possible asymptotic behaviour of solutions includes also infinite-time blowup.

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