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 $\mathbb{ R}^n\,\,(\hbox{ where }n \geqslant 4)$ and real hyperbolic space $\mathbb{ H}^n\,\, (\hbox{ where }n \geqslant 2)$. We work in framework of scritical spaces such as on weak-Lorentz space $L^{ \frac{ n}{ 2},\infty}(\mathbb{ R}^n)$ 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 $\mathbb{ H}^n$.
Categories: math.AP math.DG math.DS math.FA