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

Abstract

This paper deals with a Keller-Segel type parabolic-elliptic system involving nonlinear diffusion and chemotaxis u(t) = Delta(gamma(v)u), 0 = epsilon Delta v - v + u in a smoothly bounded domain Omega subset of R-n, n >= 1, under no-flux boundary conditions. The system contains a Fokker-Planck type diffusion with a motility function gamma(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 <= 2, k > 0 or n >= 3, k < 2/n-2. In addition, we find the conditions on parameters, k and epsilon, that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.