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

This paper focuses on investigating the time-periodic solution of the Keller–Segel system with a time-periodic source, denoted as $$f$$. Our research demonstrates the existence of a spatially heterogeneous time-periodic solution for any periodic source $$f$$. 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 $$f$$. In this paper, we removed these restrictive assumptions on $$f$$. 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.