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

• Published in: Mathematische Nachrichten Vol. 297; no. 8; pp. 3003 - 3023
• Main Authors: Loan, Nguyen Thi, Nguyen Thi, Van Anh, Van Thuy, Tran, Xuan, Pham Truong
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.08.2024
• Subjects: Asymptotic methods; dispersive estimates; Euclidean geometry; exponential stability; Hyperbolic coordinates; Keller–Segel system; Lorentz spaces; periodic solutions; polynomial stability; smoothing estimates; unconditional uniqueness; well‐posedness
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.202300311

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 Rn(wheren⩾4)$\mathbb {R}^n\,\,(\hbox{ where }n \geqslant 4)$ and real hyperbolic space Hn(wheren⩾2)$\mathbb {H}^n\,\, (\hbox{where }n \geqslant 2)$. We work in framework of critical spaces such as on weak‐Lorentz space Ln2,∞(Rn)$L^{\frac{n}{2},\infty }(\mathbb {R}^n)$ to obtain the results for the Keller–Segel system on Rn$\mathbb {R}^n$ and on Lp2(Hn)$L^{\frac{p}{2}}(\mathbb {H}^n)$ for n<p<2n$n<p<2n$ to obtain those on Hn$\mathbb {H}^n$. 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 behaviors of periodic mild solutions of the Keller–Segel system obtained in Rn$\mathbb {R}^n$ and the one in Hn$\mathbb {H}^n$.