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

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 Fujita's 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. (C) 2011 Elsevier Inc. All rights reserved.