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

TL;DR

This paper proves that solutions to the one-dimensional Keller-Segel equations with a linear sensitivity function asymptotically converge to stationary solutions, using the josiewicz-Simon gradient inequality to establish convergence.

Contribution

It demonstrates the asymptotic convergence of solutions for the Keller-Segel equations with a linear sensitivity function, extending previous results on attractors and Lyapunov functions.

Findings

Solutions converge to stationary states as time approaches infinity.

The josiewicz-Simon gradient inequality is used to prove convergence.

Global attractors exist for the equations under study.

Abstract

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.