Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis

By Yoshie Sugiyama. Differential and Integral Equations, 2007

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Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, methodology): “However, M0 will generally depend on the constant of the Gagliardo-Nirenberg-Sobolev inequality, as in [58, 59, 34].”
Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces (2009, unknown): “Moreover, in [26,27], the following decay rate was obtained ∥∥u(t)∥∥p + ∥∥v(t)∥∥p = O (t− n n(m−1)+2 (1− (1)p )), 1 p ∞, as t → ∞.”