The spatiotemporal heterogeneous time‐periodic solution of Keller–Segel system

Abstract This paper focuses on investigating the time‐periodic solution of the Keller–Segel system with a time‐periodic source, denoted as . Our research demonstrates the existence of a spatially heterogeneous time‐periodic solution for any periodic source . This study enhances and extends the results established in the work of Hieber and Stinner, where the time‐periodic solution was obtained by imposing certain smallness assumptions on . In this paper, we removed these restrictive assumptions on . Additionally, we extend the study by proving that as the chemotactic sensitivity coefficient approaches zero, the time‐periodic bacterial density tends to a spatially evenly distributed state.