Finite-Time Blowup in a Supercritical Quasilinear Parabolic-Parabolic Keller-Segel System in Dimension 2

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 (Winkler in J. Math. Pures Appl., 10.1016/j.matpur.2013.01.020, 2013) 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.