  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.

Analysis of PDEs