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Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions. (English) Zbl 1252.35087

Summary: We consider quasilinear Keller-Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system, we prove an optimal (with respect to possible nonlinear diffusions generating an explosion of solutions in finite time) finite-time blowup result. In the case of a cross-diffusion system, we give results which are optimal provided one assumes some proper non-decay condition on the nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end, we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller-Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.

MSC:

35B44 Blow-up in context of PDEs
35K55 Nonlinear parabolic equations
92C17 Cell movement (chemotaxis, etc.)
35K51 Initial-boundary value problems for second-order parabolic systems
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