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 ${\mathbb{R}}^n(\text{ where }n⩾4)$ and real hyperbolic space ${\mathbb{H}}^n(\text{where }n⩾2)$. We work in framework of scritical spaces such as on weak-Lorentz space $L^{\frac{n}{2},\mathrm{\infty }}({\mathbb{R}}^n)$ to obtain the results for Keller-Segel system on ${\mathbb{R}}^n$ and on $L^{\frac{p}{2}}({\mathbb{H}}^n)$ for $n<p<2n$ to obtain the ones on ${\mathbb{H}}^n$. 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 behaviours of periodic mild solutions of Keller-Segel system obtained in ${\mathbb{R}}^n$ and the one in ${\mathbb{H}}^n$.