  We show that the Keller-Segel model in one dimension with Neumann boundary
conditions and quadratic cellular diffusion has an intricate phase transition
diagram depending on the chemosensitivity strength. Explicit computations allow
us to find a myriad of symmetric and asymmetric stationary states whose
stability properties are mostly studied via free energy decreasing numerical
schemes. The metastability behavior and staircased free energy decay are also
illustrated via these numerical simulations.
