Periodic Solutions of the parabolic-elliptic Keller-Segel system on whole spaces

Abstract

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 ${\mathbb{R}}^n(wheren⩾4)$ and real hyperbolic space ${\mathbb{H}}^n(wheren⩾2)$. 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 ${\mathbb{H}}^n$.

arXiv comment19 pages, some errors were fixed

References in corpus (3)

• Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane
• Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth
• On asymptotically almost periodic solutions to the Navier-Stokes equations on hyperbolic manifolds

Cited by in corpus (3)

• Upper bounds for the blow-up time of the 2-D parabolic-elliptic Patlak-Keller-Segel model of chemotaxis
• Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains
• On pseudo almost Periodic Solutions of the parabolic-elliptic Keller-Segel systems