Asymptotic Convergence of Solutions for One-Dimensional Keller-Segel Equations

The second and third authors of this paper have constructed in [14] finite-dimensional attractors for the one-dimensional Keller-Segel equations. They have also remarked in [14, Section 7] that, when the sensitivity function is a linear function, the equations admit a global Lyapunov function. But at that moment they could not show the asymptotic convergence of solutions. This paper is then devoted to supplementing the results of [14, Section 7] by showing that, as $t \to \infty$, every solution necessarily converges to a stationary solution by using the {\L}ojasiewicz-Simon gradient inequality of the Lyapunov function.