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

Abstract

Copyright © 2017, arXiv, All rights reserved. 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 αϵ (0, 2). We prove uniform in time boundedness of its solution in the supercritical range α > d (1 . c), where c is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for ku(t)- u∞→0 , where u1 1 is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.