New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models

Abstract

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 some 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 jump probability 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 some initial data with mass exceeding m⁎ solutions are unbounded, though being defined for any time t>0.