We study the well-posedness and exponential stability of periodic and almost periodic mild solutions for the parabolic–parabolic Keller–Segel systems on the real hyperbolic spaces \(\mathbb {H}^d(\mathbb {R})\) , where \(d\geqslant 2\) . First, we use the dispersive estimates of the scalar heat semigroup to prove the well-posedness of bounded mild solutions for the corresponding linear systems. Then, we combine the well-posedness of the linear systems and fixed-point arguments to obtain the well-posedness of mild solutions for the semilinear systems. Moreover, we establish the exponential stability of the obtained solutions using a Gronwall-type inequality. Moreover, we give an application of the stability as a construction of periodic mild solutions for Keller–Segel systems. Finally, we prove also the existence and exponential stability of almost periodic solutions.
