Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space $R^n(\text{where}n⩾4)$ and real hyperbolic space $H^n(\text{where}n⩾2)$. We work in framework of critical spaces such as on weak-Lorentz space $L^{\frac{n}{2},\infty }\left(R^n\right)$ to obtain the results for the Keller–Segel system on $R^n$ and on $L^{\frac{p}{2}}\left(H^n\right)$ for $n<p<2n$ to obtain those on $H^n$. Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in $R^n$ and the one in $H^n$.