Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model

The asymptotic behavior of the attraction-repulsion Keller-Segel model in one dimension is studied in this paper. The global existence of classical solutions and nonconstant stationary solutions of the attraction-repulsion Keller-Segel model in one dimension were previously established by Liu and Wang (2012), which, however, only provided a time-dependent bound for solutions. In this paper, we improve the results of Liu and Wang (2012) by deriving a uniform-in-time bound for solutions and furthermore prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates.