Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction

The long-time behavior of solutions to an attraction repulsion chemotaxis system is considered in this paper in a bounded domain under zero-flux boundary conditions if repulsion dominates over attraction. It is known that under the assumption that repulsion balances/overbalances attraction in two or higher dimensions, for any suitably regular initial data all solutions of this system will be global and bounded. The present work further shows that if the degradation rate of a repulsive signal is smaller than the attractive one or the repulsion is suitably strong, this global solutions converge to the steady state for arbitrarily large initial data. To the best of our knowledge, these are the first results on the large time of large-data solutions in a higher-dimensional attraction-repulsion chemotaxis system.