Boundedness of the attraction-repulsion Keller-Segel system

This paper considers the initial-boundary value problem of the attraction-repulsion Keller-Segel model describing aggregation of Microglia in the central nervous system in Alzheimer's disease due to the interaction of chemoattractant and chemorepellent. If repulsion dominates over attraction, we show the global existence of classical solution in two dimensions and weak solution in three dimensions with large initial data.