Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type with small data

Abstract

This paper deals with the quasilinear degenerate Keller–Segel system (KS) of “parabolic–parabolic” type. The global existence of weak solutions to (KS) with small initial data is established when q ⩾ m + 2 N ( m denotes the intensity of diffusion and q denotes the nonlinearity). In the system of “parabolic–elliptic” type, Sugiyama and Kunii (2006) [13, Theorem 3] and Sugiyama (2007) [12, Theorem 2] state the similar result; note that q = m + 2 N corresponds to generalized Fujitaʼs critical exponent. However, the super-critical case where q ⩾ m + 2 N has been unsolved for “parabolic–parabolic” type. Therefore this paper gives an answer to the unsolved problem.