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 epsilon (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 vertical bar vertical bar u(t) - u(infinity)vertical bar vertical bar L-infinity -> 0, where u(infinity) 1 is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.