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.