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 $\mathbb{R}^n\,\,(\hbox{ where }n \geqslant 4)$ 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 $\mathbb{H}^n$.