Asymptotically Almost Periodic Solutions of a Chemotaxis Model on Bounded Domains

We study the existence, uniqueness, and exponential decay of asymptotically almost periodic (AAP) mild solutions to parabolic-parabolic Keller–Segel systems on a bounded domain $ {\Omega\subset 𝕉^{n}} $ with smooth boundary. First, we establish the well-posedness of mild solutions to the associated linear systems by employing dispersive and smoothing estimates for the Neumann heat semigroup on $ \Omega $ . We then prove the existence and uniqueness of AAP mild solutions to the linear systems by establishing a Massera-type principle. Next, based on the linear theory and fixed-point arguments, we derive the well-posedness of such solutions to the Keller–Segel systems. Finally, the exponential decay of these solutions is obtained by means of a Gronwall-type inequality.