Title: Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations,
Abstract: 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 $\alpha>d\left(1-c\right)$, where $c$ is an explicit
constant depending on parameters of our problem. Furthermore, we establish
sufficient conditions for $\|u(t)-u_\infty\|_{L^\infty}\rightarrow0$, where
$u_\infty\equiv 1$ is the only nontrivial homogeneous solution. Finally, we
provide a uniqueness result.