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

Abstract

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 W1, p(Ω⊂ℝn), 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 W1, p, p⩾2. Copyright © 2009 John Wiley & Sons, Ltd.