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 Łojasiewicz-Simon gradient inequality of the Lyapunov function.
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