Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing

This paper deals with a Keller-Segel type parabolic-elliptic system involving nonlinear diffusion and chemotaxis {equation presented} in a smoothly bounded domain Ω Rⁿ, n ≥ 1, under no-flux boundary conditions. The system contains a Fokker-Planck type diffusion with a motility function &gama;(v) = v^-k , k > 0. The global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when n {equation presented} In addition, we find the conditions on parameters, k and ϵ, that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.