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

Abstract This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of R N ( N ≥ 2 ). In the case of non-degenerate diffusion, Cieślak–Stinner [3] , [4] proved that if q > m + 2 N , 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 + 2 N (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 + 2 N .