Stability of steady-state solutions of a class of Keller-Segel models with mixed boundary conditions

In this paper, we investigate the existence and stability of non-trivial steady-state solutions of a class of chemotaxis models with zero-flux boundary conditions and Dirichlet boundary conditions on a one-dimensional bounded interval. By using upper-lower solution and the monotone iteration scheme method, we get the existence of the steady-state solution of the chemotaxis model. Moreover, by adopting the "inverse derivative" technique and the weighted energy method, we prove the stability of the steady-state solution of this chemotaxis model.