Abstract: In this paper we consider a dd-dimensional (d=1,2d=1,2) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order ??(0,2)??(0,2). We prove uniform in time boundedness of its solution in the supercritical range ?>d(1?c)?>d(1?c), where cc is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for ?u(t)?u??L??0?u(t)?u??L??0, where u??1u??1 is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.

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