This paper deals with a Keller-Segel type parabolic-elliptic system involving nonlinear diffusion and chemotaxisin a smoothly bounded domain Ω ⊂ R n , n 1, under no-flux boundary conditions. The system contains a Fokker-Planck type diffusion with a motility functionThe global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when n 2, k > 0 or n 3, k < 2 n−2 . In addition, we find the conditions on parameters, k and ε, that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.

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