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In this paper we consider quasilinear Keller-Segel type systems of two kindsin higher dimensions. In the case of a nonlinear diffusion system we prove anoptimal (with respect to possible nonlinear diffusions generating explosion infinite time of solutions) finite-time blowup result. In the case of across-diffusion system we give results which are optimal provided one assumessome proper non-decay of a nonlinear chemical sensitivity. Moreover, we showthat once we do not assume the above mentioned non-decay, our result cannot beas strong as in the case of nonlinear diffusion without nonlinearcross-diffusion terms. To this end we provide an example, interesting byitself, of global-in-time unbounded solutions to the nonlinear cross-diffusionKeller-Segel system with chemical sensitivity decaying fast enough, in a rangeof parameters in which there is a finite-time blowup result in a correspondingcase without nonlinear cross-diffusion.

Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions