In this paper 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 explosion in finite time of solutions) 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 of a 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.

We consider an elliptic-parabolic system of the Keller-Segel type which involves nonlinear diffusion. We find a critical exponent of the nonlinearity in the diffusion, measuring the strength of diffusion at points of high (population) densities, which distinguishes between finite-time blow-up and global-in-time existence of uniformly bounded solutions. This critical exponent depends on the space dimension n 1, where apart from the physically relevant cases n = 2 and n = 3 also the result obtained in the one-dimensional setting might be of mathematical interest: here, namely, finite-time explosion of solutions occurs although the Lyapunov functional associated with the system is bounded from below. Additionally this one-dimensional case is an example to show that L ∞ estimates of solutions to non-uniformly parabolic drift-diffusion equations cannot be expected even when boundedness of the gradient of the drift term is presupposed.

We carry on our studies related to the fully parabolic quasilinear Keller-Segel system started in [6] and continued in [7]. In the above mentioned papers we proved finite-time blowup of radially symmetric solutions to the quasilinear Keller-Segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds. On the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time. The present paper is devoted to looking for critical exponents distinguishing between those two behaviors. Moreover, we apply our results to the so-called volume filling models with a power-type probability jump function. The most important consequence of our investigations of the latter is a critical mass phenomenon found in dimension 2. Namely we find a value m * such that when the solution to the two-dimensional volume filling Keller-Segel system starts with mass smaller than m * , then it is bounded, while for initial data with mass exceeding m * solutions are unbounded, though being defined for any time t > 0.