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 in a smoothly bounded domain , , under no-flux boundary conditions. The system contains a Fokker–Planck type diffusion with a motility function , . The global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when , or , . In addition, we find the conditions on parameters, and , that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.