  In this article we investigate the existence, uniqueness and exponential
decay of asymptotically almost periodic solutions of the parabolic-elliptic
Keller-Segel system on a real hyperbolic manifold. We prove the existence and
uniqueness of such solutions in the linear equation case by using the
dispersive and smoothing estimates of the heat semigroup. Then we pass to the
well-posedness of semi-linear equation case by using the results of linear
equation and fixed point arguments. The exponential decay is proven by using
Gronwall's inequality.
