On almost periodic solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic manifold

In this work, we will study the existence, uniqueness and exponential stability of almost periodic solutions to the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We clarify the existence and uniqueness of such solutions of the linear equation by utilizing the dispersive and smoothing estimates of the heat semigroup. Thereafter, we use the fixed point arguments to investigate for the case of the semi-linear equation by utilizing the results of the linear case. Finally, we invoke the Gronwall's inequality to point out the exponential stability.