Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

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 $\mathrm{\parallel }u(t)-u_{\mathrm{\infty }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\to 0$, where $u_{\mathrm{\infty }}\equiv 1$ is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.