Burczak, Jan; Granero-Belinchón, Rafael Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. (English) Zbl 1439.35054 Discrete Contin. Dyn. Syst., Ser. S 13, No. 2, 139-164 (2020). Summary: In this paper we consider a \(d\)-dimensional \((d = 1, 2)\) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order \(\alpha \in (0, 2)\). We prove uniform in time boundedness of its solution in the supercritical range \(\alpha>d(1-c)\), where \(c\) is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for \(\|u(t)-u_\infty\|_{L^\infty}\rightarrow 0\), where \(u_\infty\equiv 1\) is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result. Cited in 8 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs 35K51 Initial-boundary value problems for second-order parabolic systems 35R11 Fractional partial differential equations 92C17 Cell movement (chemotaxis, etc.) 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35S10 Initial value problems for PDEs with pseudodifferential operators Keywords:Keller-Segel system; fractional dissipation; logistic source; global-in-time smoothness; boundedness; periodic boundary conditions; elliptic-parabolic system PDFBibTeX XMLCite \textit{J. Burczak} and \textit{R. Granero-Belinchón}, Discrete Contin. Dyn. Syst., Ser. S 13, No. 2, 139--164 (2020; Zbl 1439.35054) Full Text: DOI arXiv References: [1] P. Aceves-Sanchez and L. Cesbron, Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation, arXiv preprint, arXiv: 1606.07939, 2016. · Zbl 1516.35064 [2] P. Aceves-Sanchez and A. 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