Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type

This paper deals with the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type in a ball of RN (N≥2). In the case of non-degenerate diffusion, Cieślak-Stinner [3], [4] proved that if q>m+2N, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if q>m+2N (see Ishida-Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when q>m+2N.