The steady states and convergence to equilibria for a 1‐D chemotaxis model with volume‐filling effect

We consider a chemotaxis model with volume‐filling effect introduced by Hillen and Painter. They also proved the existence of global solutions for a compact Riemannian manifold without boundary. Moreover, the existence of a global attractor in W^{1, p}(Ω??ⁿ), p>n, p?2, was proved by Wrzosek. He also proved that the ω‐limit set consists of regular stationary solutions. In this paper, we prove that the 1‐D stationary problem has at most an infinitely countable number of regular solutions. Furthermore, we show that as t→∞ the solution of the 1‐D evolution problem converges to an equilibrium in W^{1, p}, p?2. Copyright © 2009 John Wiley & Sons, Ltd.