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Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2. (English) Zbl 1295.35123

Summary: In this note we show finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. We prove this result by slightly adapting M. Winkler’s method, which he introduced in [M. Winkler, J. Math. Pures Appl. (9) 100, No. 5, 748–767 (2013; Zbl 1326.35053)] for the semilinear Keller-Segel system in dimensions at least three, to the two-dimensional setting. This is done in the 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 non-decay assumption is essential with respect to keeping the finite-time blowup result. Namely, we prove that without the non-decay assumption solutions exist globally in time, however infinite-time blowup may occur.

MSC:

35B44 Blow-up in context of PDEs
92C17 Cell movement (chemotaxis, etc.)
35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
35B07 Axially symmetric solutions to PDEs

Citations:

Zbl 1326.35053
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References:

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